
Class r 'd Q 

Book_ 






(ipight]^". 



COPyRIGHT DEPOSIT. 



WORKS OF 
PROF. WALTER L. WEBB 



PUBLISHED BY 



JOHN WILEY & SONS. 



Railroad Construction.— Theory and Practice. 

A Text-book for the Use of Students in Colleges 
and Technical Schools. Second Edition, Reset and 
Enlarged. i6mo. xvi-\-6j6 pages and 232 figures and 
plates. Morocco, ^5.00. 
Problems in the Use and Adjustment of Engineer^ 
ing Instruments. 

Forms for Field-notes ; General Instructions for 
Extended Students' Surveys. i6mo. Morocco, 

$1.25. 



EAILROAD CONSTEUCTION. 



THEOEY AND PEACTICE, 



A TEXT-BOOK FOB THE USE OF STUDENTS 
IN COLLEGES AND TECHNICAL SCHOOLS, 



BY 

WALTER LOEING WEBB, C.E., 

Associate Member American Society of Civil Engineers; 

sometime Assistant Professor of Civil Engineering 

in the University of Pennsylvania; 

etc. 



SECOND EDITION, REVISED AND ENLARGED. 
FIRST THOUSAND. 



NEW YORK : 

JOHN WILEY & SONS. 

London : CHAPMAN & HALL, Limited. 

1903. 



t> 






THE LIBRARY OF 
CONGRESS, 


Two Copies Received 


MAR 30 1903 


Copyright Entry 


CLASS <^ XXo. No. 


COPY B. 



Copyright, 1899, 1903, 

BY 

WALTER LORING WEBB. 



iv 



PRESS OF 

BRAUNWORTH & C6. 

BOOKBINDERS AND PRINTERS 

BROOKLYN, N. Y. 



PREFACE TO FIRST EDITION. 



- The preparation of this book was begun several years ago, 
3when much of the subject-matter treated was not to be found in 
;;^rint, or was scattered through many books and pamphlets, and 
was hence unavailable for student use. Portions of the book 
have already been printed by the mimeograph process or have 
been used as lecture-notes, and hence have been subjected to 
the refining process of class-room use. 

The author would call special attention to the following 
^ features: 

a. Transition curves; the multiform-compound-curve method 
is used, which has been followed by many railroads in this 
country; the particular curves here developed have the great 
advantage of being exceedingly simple, and although the method 
is not theoretically exact, it is demonstrable that the differences 
are so small that they may safely be neglected. 

h. A system of earthwork computations by means of a slide- 
rule (which accompanies the volume) which enables one to 
compute readily the volume of the most complicated earthwork 
forms with an accuracy only limited by the precision of the 
cross-sectioning. 

c. The ^'mass curve" in earthwork; the theory and use of 
this very valuable process. 

d. Tables I, II, III, and IV have been computed ah novo. 
Tables I and II were checked (after computation) with other 
tables, which are generally considered as standard, and all 
discrepancies were further examined. They are believed to be 
perfect. 

e. Tables V, VI, VII, and IX have been borrowed, by per- 
mission, from ''Ludlow's Mathematical Tablet." It is beheved 
that five-place tables give as accurate results as actual field 

iii 



IV PREFACE TO FIRST EDITION. 

practice requires. Tables VIII and X have been compiled to 
conform wii:h Ludlow^ s system. 

The author wishes to acknowledge his indebtedness to Mr. 
Chas. A. Sims, civil engineer and railroad contractor, for reading 
and revising the portions relating to the cost of earthwork. 

Since the book is wTitten primarily for students of railroad 
engineering in technical institutions, the author has assumed 
the usual previous preparation in algebra, geometry, and trigo- 
nometry. 

Walter Loring Webb. 

University of Pennsylvania, 
Philadelphia, 
Jan. 1, 1900. 



PEEFACE TO SECOND EDITION. 



Since the issue of the first edition the author has conferred 
with many noted educators in civil engineering, among them the 
late Professors E. A. Fuertes and J. B. Johnson, regarding the 
most desirable size of page for this book. The inconvenience of 
the octavo edition for field-work was found to be limiting its use. 
It w^as therefore decided to recast the whole work and reduce 
the page from ^^ octavo" to ^' pocket-book" size. Advantage 
was then taken of the opportunity to revise freely and to add 
new matter. The original text has now been almost doubled by 
the addition of several chapters on structures, train resistance, 
rolling stock, etc., and also several chapters giving the funda- 
mental principles of the economics of railroad location. Those 
who are familiar w^th the late Mr. Wellington's masterpiece, 
''The Economic Theory of Railway Location," will readily ap- 
preciate the author's indebtedness to that work. Eut while the 
same general method has been followed, the author has taken 
advantage of the classification of operating expenses adopted 
by the Interstate Commerce Commission, has used the figures 
published by them (which were unavailable when Mr. Welling- 
ton wrote), and has developed the theory on an independent 
basis, vdth the exception of a few minor details. Those who 
deny the utility of such methods of computation are referred to 
§§ 367, 426, and elsewhere for a practical discussion of that 
subject. 

The author's primary aim has been to produce a ''text-book 
for students," and the subject-matter has therefore been cut 
down to that which may properly be required of students in 

V 



VI PREFACE TO SECOND EDITION. 

the time usually allotted to railroad work in a civil-engineering 
curriculum. On this account no extended discussion has been 
given to the multitudinous forms of various railroad devices 
in the chapters on structures. The aim has been to teach the 
principles and to guide the students into proper methods of 
investigation. 
January, 1903. 



TABLE OF CONTENTS. 



CHAPTER I. 

RAILROAD SURVEYS. 

PAGE 
Reconnoissance 1 

1. Character of a reconnoissance survey. 2. Selection of a gen- 
eral route. 3. Valley route. 4. Cross-country route. 5. Moun- 
tain route. 6. Existing maps. 7. Determination of relative 
elevations. Barometrical method. 8. Horizontal measurements, 
bearings, etc. 9, Importance of a good reconnoissance. 
Preliminary surveys 9 

10. Character of a survey. 11. Cross-section method. 12. 
Cross-sectioning. 13. Stadia method. 14. "First" and "sec- 
ond" preliminary survey. 
Location surveys 1 

15. "Paper location." 16. Surveying methods, 17. Form 
of Notes. 

CHAPTER II. 

ALIGNMENT. 

Simple curves 19 

18. Designation of curves. 19. Length of a subchord. 20. 
Length of a curve. 21. Elements of a curve. 22. Relation be- 
tween T, E, and J. 23. Elements of a 1° curve. 24. Exercises. 
25. Curve location by deflections. 26. Instrumental work. 27. 
Curve location by two transits. 28. Curve location by tangential 
offsets. 29. Curve location by middle ordinates. 30, Curve 
location by offsets from the long chord. 31. Use and value of the 
above methods. 32. Obstacles to location. 33. Modifications of 
location. 34. Limitations in location. 35. Determination of the 
curvature of existing track. 36. Problems. 

Compound curves 38 

37. Nature and use. 38. Mutual relations of the parts of a com- 
pound curve having two branches. 39. Modifications of location. 
40. Problems. 

vii 



Vlll TABLE OF CONTENTS. 



PAGE 

Transition curves 43 

41. Superelevation of the outer rail on curves. 42. Practical 
rules for superelevation. 43. Transition from level to inclined 
track. 44. Fundamental principle of transition curves. 45. 
Multiform compound curves. 46. Required length of spiral. 47. 
To find the ordinates of a l°-per-25-feet spiral. 48. To find the 
deflections from any point of the spiral. 49. Connection of spiral 
with circular curve and with tangent. 50. Field-work. 51. To 
replace a simple curve by a curve with spirals. 52. Application 
of transition curves with compound curves. 53. To replace a com- 
pound curve by a curve with spirals. 

Vertical curves 61 

54. Necessity for their use. 55. Required length. 56. Form 
of curve. 57. Numerical example. 

CHAPTER III. 

earthwork. 

Form of excavations and embankments 65 

58. Usual form of cross-section in cut and fill. 59. Terminal 
pyramids and wedges. 60. Slopes. 61. Compound sections. 
62. Width of roadbed. 63. Form of subgrade. 64. Ditches. 
65. Effect of sodding the slopes, etc. 

Earthwork surveys 73 

66. Relation of actual volume to the numerical result. 67. 
Prismoids. 68. Cross-sectioning 69. Position of slope-stakes. 
69a. Setting slope-stakes by means of "automatic" slope-stake 
rods. 

Computation of volume 79 

70. Prismoidal formula. 71. Averaging end areas. 72. Middle 
areas. 73. Two-level ground. 74. Level sections. 75. Numeri- 
cal example, level sections. 76. Equivalent sections. 77. Equiv- 
alent level sections. 78, Three-level sections. 79. Computation 
of products. 80. Five-level sections. 81. Irregular sections. 
82. Volume of an irregular prismoid. 83. True prismoid correc- 
tion for irregular prismoids. 84. Numerical example; irregular 
sections; volume, with true prismoidal correction. 85. Volume 
of irregular prismoid, with approximate prismoidal correction. 
86. Illustration of value of approximate rules. 87. Cross-section- 
ing irregular sections. 88. Side-hill work. 89. Borrow-pits. 90. 
Correction for curvature. 91. Eccentricity of the center of 
gravity. 92. Center of gravity of side-hill sections. 93. Exam- 
ples of curvature correction. 94. Accuracy of earthwork com- 
putations. 95. Approximate computations from profiles. 

Formation of embankments Ill 

96. Shrinkage of earthwork. 97. Allowance for shrinkage. 
98. Methods of forming embankments. 

Computation of haul 1 1€ 

99. Nature of subject. 100. Mass diagram. 101. Properties 



124 



139 



TABLE OF CONTENTS. IX 

PAGE 

of the mass curve. 102. Area of the mass curve. 103. Value of 
the mass diagram. 104. Changing the grade line. 105. Limit of 

free haul. 

Elements of the cost of earthwork 

106. General divisions of the subject. 107. Loosening. 108. 
Loading. 109. Hauling. 110. Choice of method of haul depend- 
ent on distance. 111. Spreading. 112. Keeping roadways in 
order. 113. Repairs, wear, depreciation, and interest on cost of 
plant. 114. Superintendence and incidentals. 115. Contractor's 
profit. 116. Limit of profitable haul. 

Blasting 

117. Explosives. 118. Drilling. 119. Position and direction 
of drill-holes. 120. Amount of explosive. 121. Tamping. 122. 
Exploding the charge. 123. Cost. 124. Classification of ex- 
cavated material. 125. Specifications for earthwork. 

CHAPTER IV. 

TRESTLES. 

126. Extent of use. 127. Trestles vs. embankments. 128. Two 

principal t5T)es 1 ^^ 

Pile trestles ^^^ 

129. Pile bents. 130. Methods of driving piles. 131. Pile- 
driving formulae. 132. Pile-points and pile-shoes. 133. Details 
of design. 134. Cost of pile trestles. 
Framed trestles ^^'^ 

135. Typical design. 136. Joints. 137. Multiple-story con- 
struction. 138. Span. 139. Foundations. 140. Longitudinal 
bracing. 141. Lateral bracing. 142. Abutments. 
Floor systems 1^^ 

143. Stringers. 144. Corbels. 145. Guard-rails. 146 Ties on 
trestles. 147. Superelevation of the outer rail on curves. 148. 
Protection from fire. 149. Timber. 150. Cost of framed timber 
trestles. 
Design of wooden trestles 169 

151. Common practice. 152. Required elements of strength. 
153. Strength of timber. 154. Loading. 155. Factors of safety. 
156. Design of stringers. 157. Design of posts. 158. Design 
of caps and sills. 159. Bracing. 

CHAPTER V. 

TUNNELS. 

Surveying 179 

150. Surface surveys. 161. Surveying down a shaft. 162. 
Underground surveys. 163. Accuracy of tunnel surveying. 

Design 184 

164. Cross-sections. 165. Grade. 166. Lining. 167. Shafts. 
168. Drains. 



TABLE OF CONTENTS, 



PAGE 

Construction 1 89 

169. Headings. 170. Enlargement. 171. Distinctive features 
of various methods of construction. 172. Ventilation during con- 
struction. 173. Excavation for the portals. 174. Tunnels vs. 
open cuts. 175. Cost of tunneling. 

CHAPTER YI. 

CULVERTS AND MINOR BRIDGES. 

176. Definition and object. 177. Elements of the design 197 

Area of the waterway 198 

178. Elements involved. 179. Methods of computation of area. 
180. Empirical formulae. 181. Value of empirical formulae. 182. 
Results based on observation, 183. Degree of accuracy required. 

Pipe culverts 202 

184. Advantages. 185. Construction. 186. Iron-pipe culverts. 
187. Tile-pipe culverts. 

Box culverts 206 

188. Wooden box culverts. 189. Stone box culverts. 190. Old 
rail culverts. 

Arch culverts 210 

191. Influence of design on flow. 192. Example of arch-cul- 
vert design. 

Minor openings 211 

193. Cattle-guards. 194. Cattle-passes. 195. Standard stringer 
and I-beam bridges. 

CHAPTER VII. 

BALLAST. 

196. Purpose and requirements. 197. Materials. 198. Cross- 
sections. 199. Methods of laying ballast. 200. Cost 215 

CHAPTER VIII. 

TIES AND OTHER FORMS OF RAIL SUPPORT. 

201. Various methods of supporting rails. 202. Economics of 

ties 220 

Wooden ties 221 

203. Choice of wood. 204. DurabiUty. 205. Dimensions. 
206. Spacing. 207. Specifications. 208. Regulations for laying 
and renewing ties. 209. Cost of ties. 
Preservative processes for wooden ties 225 

210. General principle. 211. Vulcanizing. 212. Creosoting. 
213. Burnettizing. 214. Kyanizing. 215. Wellhouse (or zinc- 
tannin) process. 216. Cost of treating. 217. Economics of 
treated ties. 



TABLE OP CONTENTS. XI 



PAGE 

Metal ties 231 

218. Extent of use. 219. Durability. 220. Form and dimen- 
sions of metal cross-ties. 221. Fastenings. 222. Cost. 223. 
Bowls or plates. 224. Longitudinals. 

CHAPTER IX. 

Hails. 

225. Early forms. 226. Present standard forms. 227. Weight 
for various kinds of traffic. 228. Effect of stiffness on traction. 
229. Length of rails. 230. Expansion of rails. 231. Rules for 
allowing for temperature, 232. Chemical composition. 233. 
Testing. 233a. Proposed standard specifications for steel rails. 
234. Rail wear on tangents. 235. Rail wear on curves. 236. 
Cost of rails 235 

CHAPTER X. 

rail-fastenings. 

Rail-joints 249 

237. Theoretical requirements for a perfect joint. 238. EflS.- 
ciency of the ordinary angle-bar. 239. Effect of rail-gap at joints. 
240. Supported, suspended, and bridge joints. 241. Failures of 
rail-joints. 242. Standard angle-bars. 243. Later designs of rail- 
joints. 243a. Proposed specifications for steel splice-bars. 

Tie-plates 255 

244. Advantages. 245. Elements of the design. 246. Methods, 
of setting. 

Spikes 258 

247. Requirements. 248. Driving. 249. Screws and bolts. 
250. Wooden spikes. |j 

Track-bolts and nut-locks 261 

251. Essential requirements. 252. Design of track-bolts. 253. 
Design of nut-locks. 

CHAPTER XI. 

switches and crossings. 

Switch construction 265 

254. Essential elements of a switch. 255. Frogs. 256. To find 
the frog number. 257. Stub switches. 258. Point switches. 259. 
Switch-stands. 260. Tie-rods. 261. Guard-rails. 

Mathematical design of switches 272 

262. Design with circular lead rails. 263. Effect of straight frog- 
rails. 264. Effect of straight point-rails. 265. Combined effect 
of straight frog-rails and straight point-rails. 266. Comparison of 
the above methods. 267. Dimensions for a turnout from the 
outer side of a curved track. 268. Dimensions for a turnout from 
the INNER side of a curved track. 269. Double turnout from a 



Xll TABLE OF CONTEIsTS. 



PAGE 

straight track. 270. Two turnouts on the same sile. 271. Con- 
necting curve from a straight track. 272. Connecting curve from 
a curved track to the outside. 273. Connecting curve from a 
curved track to the inside. 274. Crossover between two parallel 
straight tracks. 275. Crossover between two parallel curved 
tracks. 276. Practical rules for switch-laying. 

Crossings 294 

277. Two straight tracks. 278. One straight and one curved 
track. 279. Two curved tracks. 



CHAPTER XII. 

miscellaneous structures and buildings. 

Water stations and water supply 299 

280. Location. 281. Required qualities of water. 282. Tanks. 
283. Pumping. 284. Track tanks. 285. Stand pipes. 

Buildings 304 

286. Station platforms. 287. Minor stations. 288. Section 
houses. 289. Engine houses. 

Snow structures 309 

290. Snow fences. 291. Snow sheds. 
Turntables 311 



CHAPTER XIII. 
yards and terminals. 

293. Value of proper design. 294. Divisions of the subject. . . . 313 

Freight yards 314 

295. General principles, 296. Relation of yard to main tracks. 
297. Minor freight yards. 298. Transfer cranes. 299. Track 
scales. 
Engine yards 321 

300. General principles. 

Passenger terminals 323 

CHAPTER XIV. 
block signaling. 

General principles 324 

301. Two fundamental systems. 302. Manual systems. 303. 
Development of the manual systems. 304. Permissive blocking. 
305. Automatic systems. 306. Distant signals. 307. Advance 
signals. 

Mechanical details 330 

308. Signals. 309. Wires and pipes. 310. Track circuit for 
automatic signaling. 



TABLE OF CONTENTS. XIU 

CHAPTER XV. 

ROLLING STOCK. 

PAGE 

Wheels and rails 335 

311. Effect of rigidly attaching wheels to their axles. 312. 
Effect of parallel axles. 313. Effect of coning wheels. 314. 
Effect of flanging locomotive driving wheels. 315. Action of a 
locomotive pilot-truck. 

locomotives. 

General structure 342 

316. Frame. 317. Boiler. 318. Fire box. 319. Coal con- 
sumption. 320. Heating surface. 321. Loss of efficiency of 
steam pressure. 322. Tractive power. 

Running gear 353 

323. Tyr ?s of running gear. 324. Equalizing levers. 325. 
CounterbaL ncing. 326. Mutual relations of the boiler power, 
tractive power and cylinder power for various ^types. 327. Life 
of locomotives. 

CARS. 

328. Capacity and size of cars. 329. Stresses to which car- 
frames are subjected. 330. The use of metal. 331. Draft gear. 
332. Gauge of wheels and form of wheel tread 365 

train-brakes. 

333. Introduction. 334. Laws of friction as applied to this 
problem 368 

^IECHAN^SM OF BRAKES 372 

335. Hand-brakes. 336. "Straight" air brakes. 337. Auto- 
matic air brakes. 338. Tests to measure the efficiency of brakes. 
339. Brake shoes. 

CHAPTER XVI. 

TRAIN RESISTANCE. 

340. Classification of the various forms. 341. Resistances inter- 
nal to the locomotive. 342. Velocity resistances. 343. Wheel 
resistances. 344. Grade resistance. 345. Curve resistance. 346. 
Brake resistances. 347. Inertia resistance. 348. Formulae for 
train resistance. 349. Dynamometer tests. 350. Gravity or 
"drop" tests 378 

CHAPTER XVII. 

COST OF RAILROADS. 

351. General considerations. 352. Preliminary financiering. 
353. Surveys and engineering expenses. 354. Land and land 



XIY TABLE OF CONTENTS 



PAGE 

damages. 355. Clearing and grubbing. 356. Earthwork. 357. 
Bridges, trestles and culverts. 358. Trackwork. 359. Buildings 
and miscellaneous structures. 360. Interest on construction. 
361. Telegraph lines. 362. Detailed estimate of the cost of aline 
of road 393 



PART II. 

RAILROAD ECONOMICS, 
CHAPTER XVin. 

INTRODUCTION. 

36i3. The magnitude of railroad business. 364. Cost of trans- 
portation. 365. ^tudy of railroad economics — its nature and 
limitations. 366. Outline of the engineer's duties. 367. Justi- 
fication of such methods of computation 401 



CHAPTER XIX. 

THE PROMOTION OF RAILROAD PROJECTS. 

368. Method of formation of railroad corporations. 369. The 
two classes of financial interests, the security and profits of each. 

370. The small margin between profit and loss to the projectors. 

371. Extent to which a railroad is a monopoly. 372. Profit 
resulting from an increase in business done; loss resulting from a 
decrease. 373. Estimation of probable volume of traffic, and of 
probable growth. 374. Probable number of train,s per day. In- 
crease with growth of traffic, 375. Effect on traffic of an increase 
in facilities. 376. Loss caused by inconvenient terminals and 
by stations far removed from business centres. 377. General 
principles which should govern the expenditure of money for 
railroad purposes 406 

CHAPTER XX. 

OPERATING EXPENSES. 

378. Distribution of gross revenue. 379. Fourfold distribution 
of operating expenses. 380. Operating expenses per train mile. 
381. Reasons for uniformity in expenses per train mile. 382. 
Detailed classification of expenses with ratios to the total expense. 
383. Elements of the cost (with variations and tendencies) of the 
various items 419 



TABLE OF CONTENTS. XV 



PAGE 

Maintenance of way 424 

384. Item 1. Repairs of roadway. 385. Item 2. Renewal of 
rails. 386. Item 3. Renewal of ties, 387. Item 4. Repairs 
and renewals of bridges and cuWerts. 388. Items 5 to 10. Re- 
pairs and renewals of fences, road crossings, and cattle guards — 
of buildings and fixtures — of docks and wharves — of telegraph 
plant; stationery and printing; and "other expenses.'' 

Maintenance of equipment 427 

389. Item 11. Superintendence. 390. Item 12. Repairs and 
renewals of locomotives. 391. Items 13, 14, and 15. Repairs and 
renewals of passenger cars, of freight cars, and of work cars. 392. 
Items 16, 17, 18, and 19. Repairs and renewals of marine equip- 
ment, of shop machinery and tools; stationery and printing; other 
expenses. 

Conducting transportation 248 

393. Item 20. Superintendence. 394. Item 21. Engine and 
roundhouse men. 395. Item 22. Fuel for locomotives. 396. 
Items 23, 24, and 25. Water supply; oil, tallow, and waste; other 
supplies for locomotives. 397. Item 26. Train service. 398. 
Item 27. Train supplies and expenses. 399. Items 28, 29, 30, and 
31. Switchmen, flagmen, and watchmen, telegraph expenses; 
station service; and station supplies. 400. Items 32, 33, and 34. 
Switching service — balance ; car mileage — balance ; hire of equip- 
ment. 401. Items 35, 36, and 37. Loss and damage; injuries to 
persons; clearing wrecks. 402. Items 38 to 53. 



CHAPTER XXI. 

DISTANCE. 

403. Relation of distance to rates and expenses. 404. The 
conditions other than distance that affect the cost; reasons why 

rates are usually based on distance 432 

Effect of distance on operating expenses 433 

405. Effect of slight changes in distance on maintenance of way. 
406. Effect on maintenance of equipment. 407. Effect on con- 
ducting transportation. 408. Estimate of total effect on expenses 
of small changes in distance (measured in feet); estimate for dis- 
tances measured in miles. 
Effect of distance on receipts 44q 

409. Classification of traffic. 410. Method of division of through 
rates between the roads run over. 411. Effect of a change in the 
length of the home road on its receipts from through competitive 
traffic. 412. The most advantageous conditions for roads forming 
part of a through competitive route. 413. Effect of the variations 
in the length of haul and the classes of the business actually done. 
414. General conclusions regarding, a change in distance. 415. 
Justification of decreasing distance to save time. 416. Effect of 
change of distance on the business done. 



XVI TABLE OF CONTENTS. 

CHAPTER XXII. 

CURVATURE. 

PAGE 

417. General objections to curvature. 418. Financial value of 
the danger of accident due to curvature. 419. Effect of curvature 

on travel. 420. Effect on operation of trains 445 

Effect of curvature on operating expenses 449 

421. Relation of radius of curvature and of degrees of central 
angle to operating expenses. 422. Effect of curvature on mainte- 
nance of way. 423. Effect of curvature on maintenance of equip- 
ment, 424. Effect of curvature on conducting transportation. 
425. Estimate of total effect per degree of central angle. 426. 
Reliability and value of the above estimate. 
Compensation for curvature 455 

427. Reasons for compensation. 428. The proper rate of com- 
pensation. 429. The limitations of maximum curvature. 

CHAPTER XXIII. 

grade. 

430. Two distinct effects of grade. 431. Application to the 
movement of trains of the laws of accelerated motion. 432. Con- 
struction of a virtual profile. 433. Use value and possible misuse. 
434. Undulatory grades; advantages, disadvantages, and safe 
limits 460 

Minor grades 467 

435. Basis of cost of minor grades. 436. Classification of minor 
grades. 437. Effect on operating expenses. 438. Estimate of 
the cost of one foot of change of elevation. 439. Operating value 
of the removal of a hump in a grade. 

Ruling grades 473 

440. Definition. 441. Choice of ruling grades. 442. Maximum 
train load on any grade. 443. Proportion of traffic affected by 
the ruling grade. 444. Financial value of increasing the train 
load. 445. Operating value of a reduction in the rate of the ruling 
grade. 

Pusher grades 481 

446. General principles underlying the use of pusher engines. 
447. Balance of grades for pusher service. 448. Operation of 
pusher engines. 449. Length of a pusher grade. 450. Cost of 
pusher engine service, 451. Numerical comparison of pusher and 
through grades. 

Balance of grades for unequal traffic 489 

452. Nature of the subject. 453. Computation of the theoreti- 
cal balance. 454. Computation of relative traffic. 



TABLE OF CONTENTS. XVll 

CHAPTER XXIV. 

THE IMPROVEMENT OF OLD LINES. 

PAGE 

455. Classification of improvements. 456. Advantages of re- 
locations. 457. Disadvantages of re-locations 493 

Reduction of virtual grade 496 

458. Obtaining data for computations. 459. Use of the data 
obtained. 460. Reducing the starting grade at stations. 



Appendix. The adjustments of instruments 501 



Tables. 

I. Radii of curves 512 

II. Tangents, external distances and long chords for a 1** curve. . 516 

III. Switch leads ^and distances 519 

IV. Transition curves 520 

V. Logarithms of numbers 523 

VI. Logarithmic sines and tangents of small angles 543 

VII. Logarithmic sines, cosines, tangents, and cotangents 546 

VIII. Logarithmic versed sines and external secants 591 

IX. Natural sines, cosines, tangents and cotangents 637 

X. Natural versed sines and external secants 642 

XI. Reduction of barometer reading to 32*^ F 647 

XII. Barometric elevations 648 

XIII. Coefficients for corrections for temperature and humidity. . . 648 

XIV. Capacity of cylindrical water-tanks in United States standard 

gallons of 231 cubic inches 301 

XV. Number of cross ties per .mile 396 

XVI. Tons per mile (with cost) of rails of various weights 397 

XVII. Splice bars and bolts per mile of track 398 

XVIII. Railroad spikes 398 

XIX. Track bolts 398 

XX. Classification of operating expenses of aU railroads 425 

XXI. Effect on operating expenses of changes in distance 439 

XXII. Effect on operating expenses of changes in curvature 453 

XXIII. Velocity head of trains 463 

XXIV. Effect on operating expenses of changes in grade 472 

XXV. Tractive power of locomotives 475 

XXVI. Total train resistance per ton on various grades 477 

XXVII. Cost of an additional train to handle a given traffic 480 

XXVIII. Balanced grades for one, twog and three engines 485 

XXIX. Cost per mile of a pusher engine 488 

XXX. Useful trigonometrical formulae 649 

XXXI. Useful formulae and constants 651 

Index 653 



RAILROAD CONSTRUCTION. 



CHAPTER I. 

RAILROAD SURVEYS, 

The proper conduct of railroad surveys presupposes an 
adequate knowledge of almost the whole subject of railioad 
engineer mg, and particularly of some of the complicated ques- 
tions of Railroad Economics, which are not generally studied 
except at the latter part of a course in railroad engineering, if 
at all. This chapter will therefore be chiefly devoted to methods 
of instrumental work, and the problem of choosing a general 
route will be considered only as it is influenced by the topog- 
raphy or by the application of those elementary principles of 
Railroad Economics which are self-evident or which may be 
accepted by the student until he has had an opportunity of 
studying those principles in detail 

RECONNOISSANCE SURVEYS. 

1. Character of a reconnoissance survey. A reconnoissance 
survey is a very hasty examination of a belt of country to de- 
termine which of all possible or suggested routes is the most 
promising and best worthy of a more detailed survey.. It is 
essentially very rough and rapid. It aims to discover those 
salient features which instantly stamp one route as distinctly 
superior to another and so narrow the choice to routes which 
are so nearly equal in value that a more detailed survey is nec- 
essary to decide between them. 

2. Selection of a general route. The general question of 
running a railroad between two towns is usually a financial rather 



2 RAILROAD CONSTRUCTIOX. § 3. 

than an engineering question. Financial considerations usually 
determine that a road must pass through certain more or less 
important towns between its termini. When a railroad runs 
through a thickly settled and very flat countr}^ where, from a 
topographical standpoint, the road may be run by any desired 
route, the "right-of-way agenf sometimes has a greater influ- 
ence in locating the road than the engineer. But such modifi- 
cations of alignment, on account of business considerations, are 
foreign to the engineer's side of the subject, and it will be here- 
after assumed that topography alone determines the location of 
the line. The consideration of those larger questions combin- 
ing finance and engineering (such as passing by a town on ac- 
count of the necessary introduction of heavy grades in order to 
reach it) will be taken up in Chap. XIX, et seq. 

3. Valley route. This is perhaps the simplest problem. If 
the two towms to be connected lie in the same valley, it is fre- 
quently only necessary to run a line which shall have a nearly 
uniform grade. The reconnoissance problem consists largely in 
determining the difference of elevation of the two termini of 
this division and the approximate horizontal distance so that the 
proper grade may be chosen. If there is a large river running 
through the valley, the road will probably remain on one side 
or the other throughout the whole distance, and both banks 
should be examined by the reconnoissance party to determine 
which is preferable. If the river may be easily bridged, both 
banks may be alternately used, especially when better alignment 
is thereby secured. A river valley has usually a steeper slope 
in the upper part than in the lower part. A uniform grade 
throughout the valley will therefore require that the road climbs 
up the side slopes in the lower part of the valley. In case the 
'^ruling grade" * for the whole road is as great as or greater 
than the steepest natural valley slope, more freedom may be 
used in adopting that alignment which has the least cost — 
regardless of grade. The natural slope of large rivers is almost 
invariably so low that grade has no influence in determining the 
choice of location. When bridging is necessary, the river 
banks should be examined for suitable locations for abutments 



* The ruling grade may here be loosely defined as the maximum grade 
which is permissible. This definition is not strictly true, as may be seen later 
when studying Railroad Economics, but it may here serve the purpose. 



§ 4. RAILROAD SURVEYS. 3 

and piers. If the soil is soft and treacherous, much difficulty 
ma\^ be experienced and the choice of route may be largely 
determined by the difficulty of bridging the river except at 
certain favora]:)le places. 

4. Cross-country route. A cross-country route always has one 
or more summits to be crossed. The problem becomes more 
complex on account of the greater number of possible solutions 
and the difficulty of properly weighing the advantages and dis- 
advantages of each. The general aim should be to choose the 
lowest summits and the highest stream crossings, provided that 
by so doing the grades between these determining points shall 
be as low as possible iand shall not be greater than the ruling 
grade of the road. Nearly all railroads combine cross-country 
and valle}^ routes to some extent. Usually the steepest natural 
slopes are to be found on the cross-country routes, and also the 
greatest difficulty in securing a low through grade. An approx- 
imate determination of the ruling grade is usually made during 
the reconnoissance. If the ruling grade has been pre^^ously 
decided on by other considerations, the leading feature of the 
reconnoissance survey will be the determination of a general 
route along which it will be possible to survey a line whose 
maximum grade shall not exceed the ruling grade. 

5. Mountain route. The streams of a mountainous region 
frequently have a slope exceeding the desired ruling grade. In 
such cases there is no possibility of securing the desired grade 
by following the streams. The penetration of such a region 
may only be accomplished by '' development^' — accompanied 
perhaps by tunneling. ^'Development" consists in deliber- 
ately increasing the length of the road between two extremes 
of elevation so that the rate of grade shall be as low as desired. 
The usual method of accomplishing this is to take advantage of 
some convenient formation of the ground to introduce some 
lateral deviation. The methods may be somewhat classified as 
follows : 

(a) Running the line up a convenient lateral valley, turning 
a sharp curve and working back up the opposite slope. As 
shoAMi in Fig. 1, the considerable rise between A and B was 
surmounted by starting off in a very different direction from 
the general direction of the road; then, when about one-half of 
the desired rise had been obtained, the line crossed the valley 
and continued the climb along the opposite slope, (b) Switch' 



RAILROAD CONSTRUCTION. 



§5. 



hack. On the steep side-hill BCD (Fig. 1) a very considerable 
gain in elevation was accomplished by the switchback CD. 
The gain in elevation from 5 to D is very great. On the other 
hand, the speed must always be slow; there are two complete 
stoppages of the train for each run; all trains must run back- 
ward from C to D. (c) Bridge spiral. When a valley is so 
narrow at some point that a bridge or viaduct of reasonable 
length can span the valley at a considerable elevation above the 




Fig. 1. 

bottom of the valley, a bridge spiral may be desirable. In Fig. 2 
the line ascends the stream valley past .4, crosses the stream at 
B, works back to the narrow place at C, and there crosses itself, 
having gained perhaps 100 feet in elevation, (d) Tunnel 
spiral (Fig. 3). This is the reverse of the previous plan. It 
implies a thin steep ridge, so thin at some place that a tunnel 
through it will not be excessively long. Switchbacks and 
spirals are sometimes necessary in mountainous countries, but 
they should not be considered as normal types of construction. 
A region must be very difiicult if these devices cannot be avoided. 
On Plate I are shoAvn three separate ways (as actuall}^ con- 
structed) of running a railroad between two points a little over 
three miles apart and having a difference of elevation of nearly 




Ti. 




(To face page A.) 



§5. 



RAILROAD SURVEYS. 



1100 feet. At A the Central R. R. of New Jersey runs under 
the Lehigh Valley R. R. and soon turns off to the northeast for 
about six miles, then doubles back, reaching D, a fall of about 
1050 feet with a track distance of about 12.7 miles. The 
L. V. R. R. at A runs to the westward for six to seven miles, 








Fig. 2. 



Fig. 3. 



then turns back until the roads are again close together at D, 
The track distance is about 14 miles and the drop a little greater, 
since at A the L. V. R. R. crosses over the other, while at D they 
are at practically the same level. From B to C the distance is 
over eleven miles. From A directly down to D the C. R. R. of 
N. J. runs a '' gravity" road, used exclusively for freight, on 
which cars alone are hauled by cable. The main-line routes 
are remarkable examples of sheer "development." Even as 
constructed the L. V. R. R. has a grade of about 95 feet per 
mile, and this grade has proved so excessive for freight work 
that the company has constructed a cut-off (not sho^^no. on the 
map) which leaves the main line at A, nearly parallels the 
C. R. R. to C, and then running in a northeasterly direction 
again joins the main line beyond Wilkesbarre. The grade is 
thereby cut down to 65 feet per mile. 

Rack railways and cable roads, although types of moimtain 
railroad construction, will not be here considered. 



6 



KAILROAD CONSTRUCTION, 



§6. 



6. Existing maps. The maps of the U. S. Geological Survey 
are exceedingly valuable as far as they have been completed. 
So far as topographical considerations are concerned, they 
almost dispense with the necessity for the reconnoissance and 
''first preliminary" surveys. Some of the State Survey maps 
will give practically the same information. County and town- 
ship maps can often be used for considerable information as to the 
relative horizontal position of governing points, and even some 
approximate data regarding elevations may be obtained by a 
study of the streams. Of course such information will not dis- 
pense with surveys, but will assist in so planning them as to 
obtain the best information with the least work. When the 
relative horizontal positions of points are reliably indicated on a 
map, the reconnoissance may be reduced to the determination 
of the relative elevations of the governing points of the route. 

7. Determination of relative elevations. A recent description 
of European methods includes spirit-leveling in the reconnois- 
sance work. This may be due to the fact that, as indicated 
above, previous topographical surveys have rendered unnecessary 
the '' exploratory'' survey which is required in a new country, 
and that their reconnoissance really corresponds more nearly to 
our preliminary. 

The perfection to which barometrical methods have been 
brought has rendered it possible to determine differences of 
elevation with sufficient accuracy for reconnoissance purposes 
by the combined use of a mercurial and an aneroid barometer. 
The mercurial barometer should be kept at "headquarters," and 
readings should be taken on it at such frequent intervals that 
any fluctuation is noted, and throughout the period that observa- 
tions with the aneroid are taken in the field. At each observa- 
tion there should also be recorded the time, the reading of the 
attached thermometer, and the temperature of the external 
air. For uniformity, the mercurial readings should then be 
'^ reduced to 32° F." The form of notes for the mercurial 
barometer readings should be as follows : 



Time. 


Merc. 
Barom. 


Attached 
Therm. 


Reduction 
to 32° F. 


External 
Therm. 


Corrected 
reading. 


7:00 A.M. 
:15 
:30 
:45 


29.872 
.866 
.858 
.850 


72° 

73.5 

75 

76 


— .117 
.121 
. ] 25 
.127 


73° 
75 

76 

77 


29.755 
.745 
.733 
.723 



§ 7. RAILROAD SURVEYS. 7 

The corrections in column 4 are derived from Table XI by 
interpolation. 

Before starting out, a reading of the aneroid should be taken 
at headquarters coincident with a reading of the mercurial. 
The difference is one value of the correction to the aneroid. 
As soon as the aneroid is brought back another comparison of 
readings should be made. Even though there has been con- 
siderable rise or fall of pressure in the interval, the difference 
in readings (the correction) should be substantially the same 
provided the aneroid is a good instrument. If the difference 
of elevation is excessive (as when climbing a high mountain) 
even the best aneroid will ^4ag'' and not recover its normal 
reading for several hours, but this does not apply to such dif- 
ferences of elevation as are met with in railroad work. The 
best aneroids read directly to y^ of an inch of mercury and 
may be estimated to yoVo of an inch — ^which corresponds 
to about 0.9 foot difference of elevation. In the field there 
should be read, at each point whose elevation is desired, the 
aneroid, the time, and the temperature. These readings, cor- 
rected by the mean value of the correction between the aneroid 
and the mercurial, should then be combined with the reading 
of the mercurial (interpolated if necessary) for the times of 
the aneroid observations and the difference of elevation ob- 
tained. The field notes for the aneroid should be taken as 
shoT\Ti in the first four columns of the tabular form. The '' cor- 
rected aneroid'' readings of column 5 are found by correcting 
the readings of column 3 by the mean difference between the 
mercurial and aneroid w^hen compared at morning and night. 
Column 6 is a copy of the '' corrected readings" from the office 
notes, interpolated when necessary for the proper time. Column 
7 is similarly obtained. Col. 8 is obtained from cols. 4 and 5, 
and col. 9 from cols. 6 and 7, with the aid of Table XII. The 
correction for temperature (col. 11), which is generally small 
unless the difference of elevation is large, is obtained mth the 
aid of Table XIII. The elevations in Table XII are elevations 
above an assumed datum plane, where under the given atmos- 
pheric conditions the mercurial reading would be 30''. Of 
course the position of this assumed plane changes with varying 
atmospheric conditions and so the elevations are to be con- 
sidered as relative and their difference taken. [See the author's 
''Problems in the Use and Adjustment of Engin3ering In- 



8 



RAILROAD CONSTRUCTION, 



§8, 



(Left-hand page of Notes.) 



Time. 


Place. 


Aneroid. 


Therm. 


Corr. 
Aner. 


Corr. 
Merc. 


7:00 


Office 

/lo 

saddle-back 
river cross. 


29.628 
29.662 
29.374 
29.548 


73° 
72° 
63° 
70° 




29.755 


7:10 
7:30 
7:50 


29.789 
29.501 
29.675 


29.748 
29.733 
29.720 



struments/' Prob. 22.] Important points should be observed 
more than once if possible. Such duplicate observations will be 
found to give surprisingly concordant results even when a 
general fluctuation of atmospheric pressure so modifies the 
tabulated readings that an agreement is not at first apparent. 
Variations of pressure produced by high winds, thunder-storms, 
etc., will generally vitiate possible accuracy by this method. 
By '^ headquarters^^ is meant any place Avhose elevation above 
any given datum is known and where the mercurial may be 
placed and observed while observations vathin a range of several 
miles are made with the aneroid. If necessary, the elevation of 
a new headquarters may be determined by the above method, 
but there should be if possible several independent observations 
w^hose accordance will give a fair idea of their accuracy. 

The above method should be neither slighted nor used for 
more than it is worth. When properly used, the errors are 
compensating rather than cumulative. When used, for example, 
to tietermine that a pass B is 260 feet higher than a determined 
bridge crossing at A which is six miles distant, and that another 
pass C is 310 feet higher than A and is ten miles distant, the 
figures, even with all necessary allowances for inaccuracy, will 
give an engineer a good idea as to the choice of route especially 
as affected by ruling grade. There is no comparison between 
the time and labor involved in obtaining the above information 
by barometric and by spirit-leveling methods, and for recon- 
noissance purposes the added accuracy of the spirit-leveling 
method is hardly worth its cost. 

8. Horizontal measurements, bearings, etc. When there is 
no map which may be depended on, or when only a skeleton 
map is obtainable, a rapid survey, sufficiently accurate for the 
purpose, may be made by using a pocket compass for bearings 
and a telemeter, odometer, or pedometer for distances. The 
telemeter [stadia] is more accurate, but it requires a definite clear 



§9. 



RAILROAD SURVEYS. 



(Right-hand page of Notes.) 



Temp, at 
headqu. 


Approx. 
field read. 


Approx. 
headq. read. 


DifiF. 


Corr. for 
temp. 


Diff. 
elev. 


75° 
76 

77 


192 

457 
297 


230 
244 
256 


- 38 
+ 213 
+ 41 


-(+ 2) 
+ ( + 10) 
+ (+ 2) 


— 40 
+ 223 
+ 43 



sight from station to station, which may be difficult through a 
wooded country. The odometer, which records the revolutions 
of a wheel of known circumference, may be used even in rough 
and wooded country, and the results may be depended on to a 
small percentage. The pedometer (or pace-measurer) depends 
for its accuracy on the actual moA^ement of the mechanism for 
each pace and on the uniformity of the pacing. Its results are 
necessaril}^ rough and approximate, but it may be used to fill 
in some intermediate points in a large skeleton map. A hand- 
level is also useful in determining the relative elevation of various 
topographical features which may have some bearing on the 
proper location of the road 

9. Importance of a good reconnoissance. The foregoing in- 
struments and methods should be considered only as aids in 
exercising an educated common sense, without which a proper 
location cannot be made. The reconnoissance survey should 
command the best talent and the greatest experience available. 
If the general route is properly chosen, a comparatively low 
order of engineering skill can fill in a location which will prove 
a paying railroad property ; but if the general route is so chosen 
that the ruling grades are high and the business obtained is small 
and subject to competition, no amount of perfection in detailed 
alignment or roadbed construction can make the road a profitable 
investment. 



PRELIMINARY SUR\^YS. 

10. Character of survey. A preliminary railroad survey is 
properly a topographical survey of a belt of country which has 
been selected during the reconnoissance and within which it is 
estimated that the located line will lie. The width of this belt 
will depend on the character of the country. When a railroad 
is to follow a river having very steep banks the choice of loca- 
tion is sometimes limited at places to a very few feet of width 



10 



RAILROAD CONSTRUCTION, 



§ n 



and the belt to be surveyed may be correspondingly narrowed. 
In very flat country the desired width may be only limited by the 
ability to survey points with sufficient accuracy at a considerable 
distance from what may be called the ^'backbone line" of the 
survey. 

II. Cross-section method. This is the only feasible method 
in a wooded country, and is employed by many for all kinds 
of country. The backbone line is surveyed either by observ- 
ing magnetic bearings with a compass or by carrying forward 




Fig. 4. 



absolute azimuths w^ith a transit. The compass method nas 
the disadvantages of limited accuracy and the possibility of 
considerable local error owing to local attraction. On the other 



§ 12. RAILROAD SURVEYS. 11 

hand there are the advantages of greater simplicity, no necessity 
for a back rodman, and the fact that the errors are purely 
local and not cumulative, and may be so limited, with care, that 
they will cause no vital error in the subsequent location survey. 
The transit method is essentially more accurate, but is liable 
to be more laborious and troublesome. If a large tree is en- 
countered, either it must be cut down or a troublesome opera- 
tion of offsetting must be used. If the compass is employed 
under these circumstances, it need only be set up on the far side 
of the tree and the former bearing produced. An error in 
reading a transit azimuth will be carried on throughout the 
survey. An error of only five minutes of arc will cause an off- 
set of nearly eight feet in a mile. Large azimuth errors may, 
however, be avoided by immediately checking each new azimuth 
with a needle reading. It is advisable to obtain true azimuth 
at the beginning of the survey by an observation on the sun or 
Polaris, and to check the azimuths every few miles by azimuth 
observations. Distances along the backbone line should be 
measured with a chain or steel tape and stakes set every 100 
feet. When a course ends at a substation, as is usually the case, 
the remaining portion of the 100 feet should be measured along 
the next course. The level party should immediately obtain the 
elevations (to the nearest tenth of a foot) of all stations, and also 
of the lowest points of all streams crossed and even of dry gullies 
which would require culverts. 

12. Cross-sec tionmg. It is usually desirable to obtain con- 
tours at five-foot intervals This may readily be done by the 
use of a Locke level (which should be held on top of a simple 
five-foot stick), a tape, and a rod ten feet in length graduated 
to feet and tenths. The method of use may perhaps be best 
explained by an example. Let Fig. 5 represent a section per- 
pendicular to the survey line — such a section as would be made 
by the dotted lines in Fig. 4. C represents the station point. 
Its elevation as determined by the level is, say, 158.3 above 
datum. When the Locke level on its five-foot rod is placed at 
C, the level has an elevation of 163.3. Therefore when a point 
is found (as at a) w^here the level will read 3.3 on the rod, that 
point has an elevation of 160.0 and its distance from the center 
gives the position of the 160-foot contour. Leaving the long 
rod at that point (a), carry the level to some point (6) such that 
the level will sight at the top of the rod. h is then on the 165- 



12 



RAILROAD CONSTRUCTION, 



§12. 



foot contour, and the horizontal distance ah added to the hori- 
zontal distance ac gives the position of that contour from the 
center. The contours on the lower side are found similarly. 
The first rod reading will be 8.3, giving the 155-foot contour. 




Fig. 5. 



Plot the results in a note-book which is ruled in quarter-inch 
squares, using a scale of 100 feet per inch in both directions. 
Plot the work up the page; then when looking ahead along the 
line, the work is properly oriented. When a contour crosses 



— -1.- 




Fig. 6. 

the survey line, the place of crossing may be similarly deter- 
mined. If the ground flattens out so that five-foot contours are 
very far apart, the absolute elevations of points at even fifty- 



§ 13. RAILROAD SURVEYS. 13 

foot distances from the renter should be determined. The 
method m exceedingly rapid. Whatever error or inaccuracy- 
occurs IS confined in its effect to the one station Avhere it occurs. 
The work being thus plotted in the field, \inusuallv irregular 
topography may l)e plotted with greater (-ertainty and no gK-at 
error can occur without detection, it would even be possible 
by this method to detect a gross error that might have been 
made by the le\el party 

13. Stadia method. This method is best adapted to fairly 
open country where a ^\shot" to any desired point may be 
taken without clearing. The backbone sur\'e\ hne is the same 
as in the previous method except that each course is limited to 
the practicable length of a stadia sight. The distance betvveen 
stations should be checked by foiesight and backsight— also the 
vertical angle. Azimuths should be checked by the needle. 
Considering the vital importance of leAeling on a railroad surA'ey 
it might be considered desirable to run a line of levels over the 
stadia stations m order that the leveling may be as precise as 
possible; but when it is considered that a preliminary survey is 
a somewhat hasty survey of a route that rnat/ be abandoned, and 
that the errors of leveling by the stadia method (which are con- 
pensating) may be so minimized that no proposed route would 
be abandoned on account of such small error, and that the effect 
of such an error may be easily neutralized by a slight change in 
the location, it may be seen that excessive care in the leveling 
of the preliminary survey is hardly justifiable. 

Since the students taking this work are assumed to be familiar 
with the methods of stadia topographical surveys, this part of 
the subject will not be further elaborated. 

14. " First " and " Second '* preliminary surveys. Some engi- 
neers advocate two preliminary surve}'s. When this is done, 
the first IS a very rapid survey, made perhaps with a compass, 
ard IS only a better grade of reconnoissance. Its aim is to 
rap'dlv develop the facts which will decide for or against any- 
proposed route, so that if a route is found to be unfavorable 
another more or less modified route may be adopted without 
having wasted considerable time in the survey of useless details. 
By this time the student should have grasped the fundamental 
idea that l)Oth the reconnoissance and preliminary^ surveys are 
not surveys of lines but of areas, that their aim is to survey 
only those topographical features which would have a deter- 



14 RAILROAD CONSTRUCTION. § 15. 

mining influence on any railroad line whick might be constructed 
through that particular territory, and that the value of a locating 
engineer is largely measured by his ability to recognize those 
determining influences with the least amount of work from his 
surveying corps. Frequently toa .little time is spent on the 
comparative study of preliminary lilies. A lirie will be hastily 
decided on after very little study ; it wall then be surveyed with 
minute detail and estimates carefully worked up, and the claims 
of any other suggested route will then be handicapped, if not 
disregarded, owing to an unwillingness to discredit and throw 
away a large amount of detailed surveying. The cost of two or 
three extra preliminary surveys {at critical sections and not over 
the whole line) is utterly insignificant compared with the prob- 
able improvement in the ''operating value" of a line located 
after such a comparative study of preliminary lines. 

LOCATION SURVEYS. 

15. "Paper location." When the preliminary survey has 
been plotted to a scale of 200 feet per inch and the contours 
drawn in, a study may be made for the location survey. Disre- 
garding for the present the effect on location of transition curves, 
the alignment may be said to consist of straight lines (or ''tan- 
gents") and circular curves. The "paper location" therefore 
consists in plotting on the preliminary map a succession of 
straight lines which are tangent to the circular curves connect- 
ing them. The determining points should first be considered. 
Such points are the termini of the road, the lowest practicable 
point over a summit, a river-crossing, etc. So far as is possi- 
ble, having due regard to other considerations, the road should 
be a "surface" road, i.e., the cut and fill should be made as 
small as possible. The maximum permissible grade must also 
have been determined and duly considered. The method of 
location differs radically according as the lines joining the deter- 
mining points have a very low grade or have a grade that ap- 
proaches the maximum permissible. With very low natural 
grades it is only necessar}^ to strike a proper balance between 
the requirements for easy alignment and the avoidance of exces- 
sive earthvv^ork. When the grade betw^een two determined 
points approaches the maximum, a study of the location may be 
begun by finding a strictly surface line which wiU connect those 



§ 16. RAILROAD SURVEYS. 15 

points with a line at the given grade. For example, suppose 
the required grade is 1.6% and that the contours are drawn at 
5- foot intervals 'It wjII require 312 feet of 1.6% grade to rise 
5 feet. Set a pair of dividers at 312 feet and step ofT this in- 
terval on successive contours. This line will in general be very 
irregular, but m an easy country it may lie fairly close to the 
proper location line, and even in difficult country such a surface 
line will assist greatly in selecting a suitable location. When the 
larger part of the line will evidently consist of tangents, the tan- 
gents should be first located and should then be connected by 
suitable curves. When the curves predominate, as they gener- 
ally will in mountainous country, and particularly w^hen the line 
is purposely lengthened in order to reduce the grade, the curves 
should be plotted first and the tangents ma}^ then be drawn 
connecting them. Considering the ease with w^hich such lines 
may be draw^n on the preliminary map, it is frequently advisable, 
after making such a paper location, to begin all over, draw a 
new line over some specially difficult section and compare re- 
sults. Profiles of such lines may be readily drawn by noting their 
intersection with each contour crossed. Drawing on each profile 
the required grade line will furnish an approximate idea of the 
comparative amount of earthwork required. After deciding on 
the paper location, the length of each tangent, the central angle 
(see § 21), and the radius of each curve should be measured as 
accurately as possible. Since a slight error made in such meas- 
urements, taken from a map with a scale of 200 feet per inch, 
would by accumulation cause serious discrepancies between the 
plotted location and the location as afterw^ard surveyed in the 
field, frequent tie lines and angles should be determined between 
the plotted location line and the preliminary line, and the loca- 
tion should be altered, as inay prove necessary, by changing the 
length of a tangent or changing the central angle or radius of a 
curve, so that the agreement of the check-points will be suffi- 
ciently close. The errors of an inaccurate preliminary survey 
may thus be easily neutralized (see § 33). When the pre- 
liminary hne has been properly run, its ^'backbone" line will 
lie very near the location line and will probably cross it at fre- 
quent intervals, thus rendering it easy to obtain short and nu- 
merous tie hues. 

i6. Surveying methods. A transit should be used for align- 
ment, and only precise work is allow^able. The transit stations 



16 



RAILROAD CONSTRUCTION. 



§16. 



should be centered with tacks and should be tied to witness- 
stakes, which should be located outside of the range of the earth- 
work, so that they will neither be dug up nor covered up. All 
original property lines lying within the limits of the right of way 
should be surve.ved with reference to the Jocation line, so that 
the right-of-way agent may have a proper basis for settlement. 
When the property lines do not extend far outside of the re- 
quired right of way they are frequently surveyed completely. 

The leveler usually reads the target to the nearest thousandth 
of a foot on turning-points and bench-marks, but reads to the 
nearest tenth of a foot for the elevation of the ground at stations. 
Considering that y^Vo ^^ ^ ^^^^ has an angular value of only 7 



FORM OF NOTES. 



[Left-hand page.] 










Sta. 


Align- 
ment. 


Vernier. 


Tangential 
Deflection. 


Calculated 
. Bearing. 


Needle. 


54 












53 

-h72.2 


P.T. 


9° 11' 


18° 22' 


N 54° 48' E 


N 62° 15' E 


52 




7 57 








51 
50 


1 1 


6 15 
4 33 








49 


o o 


2 51 








48 


l_Ij 


1 09 








0+32 
47 


P.C. 


0° 








46 








N 36° 26' E 


N 44° 0' E 



16. 



RAILROAD SURVEYS. 



17 



seconds at a distance of 300 feet, and that one division of a level- 
bubble is usually about 30 seconds, it ma^^ be seen that it is a 
useless refinement to read to thousandths unless corresponding 
care is taken in the use of the level. The leveler should also 
locate his bench-marks outside of the range of earthwork. A 
knob of rock protruding from the ground affords an excellent 
mark. A large nail, driven in the roots of a tree, which is not 
to be disturbed, is also a good mark. These marks should be 
clearly described in the note-book. The leveler should obtain 
the elevation of the ground at all station-points; also at all 
sudden breaks in the profile line, determining also the distance 
of these breaks from the previous even station. This will in- 



[Right-hand page.] 




53+60 
() JAS. WILSON 




wm. brown 




18 RAILROAD CONSTRUCTION. § 17. 

elude the position and elevation of all streams, and even dry 
gullies, which are crossed 

Measurements should preferably be made wich a steel tape, 
care being taken on steep ground to insure horizontal measure- 
ments. Stakes are set each 100 feet, and also at the beginning 
and end of all curves. Transit-points (sometimes called '^ plugs" 
or ^'hubs") should be driven flush with the ground, and a 
^'witness-stake," having the '' number " of the station^ should 
be set three feet to the right. For example, the witness-stake 
might have on one side "137 + 69.92/' and on the other side 
''PC4°R," which would signify that the transit hub is 69.92 
feet beyond station 137, or 13769.92 feet from the beginning of 
the line, and also that it is the '" point of curve" of a '' 4° curve" 
which turns to the right. 

Alignment. The alignment is evidently a part of the loca- 
tion survey, but, on account of the magnitude and importance 
of the subject, it will be treated in a separate chapter. 

17. Form of Notes. Although the Form of Notes cannot be 
thoroughly understood until after curves are studied, it is here 
introduced as being the most convenient place. The right-hand 
page should have a sketch showing all roads, streams, and 
property lines crossed with the bearings of those lines. This 
should be drawn to a scale of 100 feet per inch — the quarter- 
inch squares which are usually ruled in note-books giving con- 
venient 25-foot spaces. This sketch will always be more or less 
distorted on curves, since the center line is always shown as 
straight regardless of curves. The station points (^'Sta." in 
first column, left-hand page) should be placed opposite to their 
sketched positions, which means that even stations will be 
recorded on every fourth line. This allows three intermediate 
lines for substations, which is ordinarily more than sufficient. 
The notes should read up the page, so that the sketch will be 
properly oriented when looking ahead along the line The 
other columns on the left-hand page will be self-explanator}^ 
when the subject of curves is understood. If the '^ calculated 
bearings" are based on azimuthal observations, their agreement 
(or constant difference) with the needle readings will form a 
valuable check on the curve calculations and the instrumental 
work. 



CHAPTER II. 



ALIGNMENT. 



In this chapter the alignment of the center line only of a 
pair of rails is considered. When a railroad is crossing a sum- 
mit in the grade hne, although the horizontal projection of the 
ahgnment may be straight, the vertical projection will consist of 
two sloping lines joined by a curve. When a curve is on a 
grade, the center line is really a spiral, a curve of double curva- 
ture, although its horizontal projection is a circle. The center 
line therefore consists of straight lines and curves of single 
and double curvature. The simplest method of treating them 
is to consider their horizontal and vertical projections separately. 
In treating simple, compound, and transition curves, only the 
horizontal projections of those curves will be considered. 

SIMPLE CURVES. 

1 8. Designation of curves. A curve may be designated either 
by its radius or by the angle subtended by a chord of unit 
length. Such an angle is known 
as the '' degree of curve '' and is 
indicated by D. Since the curves 
that are practically used have very 
long radii, it is generally iinpracti- 
cable to make any use of the actual 
center, and the curve is located 
without reference to it. If AB in 
Fig. 4 represents a unit chord (C) 
of a curve of radius R, then b}^ the 
above definition the angle AOB 
equals D, Then 




R 



sin ^D' 



(1) 



19 



20 



RAILROAD CONSTRUCTION 



§19. 



or, by inversion, 



sinJD=2^. 



(2) 



The unit chord is variously taken throughout the world as 
100 feet, 66 feet, and 20 meters. In the United States 100 
feet is invariably used as the unit chord length, and throughout 
this work it will be so considered. Table I has been computed 
on this basis. It gives the radius, with its logarithm, of all 
curves from a 0° 01' curve up to a 10° curve, varying by single 
minutes. The sharper curves, which are seldom used, are given 
with larger intervals. 

An approximate value of R may be rea^dily found from the 
following simple rule, w^hich should be memorized: 



R^ 



5730 



D 



Although such values are not mathematically correct, since R 
does not strictly A^ary inversely as D, yet the resulting value is 
within a tenth of one per cent for all commonly used values 
of R, and is sufficiently close for many purposes, as will be 
shown later. 

19. Length of a subchord. Since it is impracticable to 
i^easure along a curved arc, curves are always measured by 

laying off lOO-foot chord lengths. 
This means that the actual arc is 
always a little longer than the 
chord. It also means that a sub- 
chord (a chord shorter than the unit 
length) will be a little loiiger than 
t)ie ratio of the angles subtended 
would call for. The truth of this 
may be seen without calculation 
by noting that two equal sub- 
chords, each subtending the angle 
Fig. $. ^-D, w^ill evidenth' be slightly longer 

than 50 feet each. If c be the length of a subchord subtend- 
ing the angle dj then, as in Eq. 2, 




sin ^d = 



2R' 



§ 20. ALIGNMENT. 2'i 

or, by inversion, 

c =272 sin i^ . (3) 

The nominal length of a subchord=100^ For example, 

ca nominal subchord of 40 feet will subtend an angle of y^^^ of 
D°; its true length will be slightly more than 40 feet, and may 
be computed by Eq. 3. The difference between the nominal 
and true lengths is maximum when the subchord is about 57 
feet long, but with the low degrees of curvature ordinarily used 
the difference may be neglected. With a 10° curve and a 
nominal chord length of 60 feet, the true length is 60.049 feet. 
A'^ery sharp curves should be laid off with 50-foot or even 25- 
foot chords (nominal length). In such cases especially the true 
lengths of these sub chords should be computed and used instead 
of the nominal lengths. 

20. Length of a curve. The length of a curve is always 
indicated by the quotient of lOOJ-^D. If the quotient of 
J^D is a whole number, the length as thus indicated is the 
true length — measured in 100-foot chord lengths. If it is an 
odd number or if the curve begins and ends with a subchord 
(even though J-=-Z) is a whole number), theoretical accuracy 
requires that the true subchord lengths shall be used, although 
the difference may prove insignificant. The length of the arc 
(or the mean length of the two rails) is therefore always in 
excess of the length as given above. Ordinarily the amount 
of this excess is of no practical importance. It simply adds an 
insignificant amount to the length of rail required. 

Example. Required the nominal and true lengths of a 
3° 45' curve having a central angle of 17° 25'. First reduce 
the degrees and minutrs to decimals of a degree. (lOOX 1 7° 25') 
--3° 45' = 1741 667^3.75=464 444. The curve has four 100- 
foot chords and a nominal chord of 64.444 The true chord" 
should be 64.451. The actual arc is 

17°.4167Xj|^X 72 =464.527 

The excess is therefore 464 527-464.451 =0.076 foot. 

21. Elements of a curve. Considering the line as running 
from A toward B, the beginning of the curve, at A, is called 
the point of curve {PC). The other end of the curve, at B, is 



22 



RAILROAD CONSTRUCTION. 



§22, 



called the 'point of tangency (PT). The intersection of the 

tangents is called the vertex (V). 
The angle made by the tangents 
at V, which equals the angle 
made by the radii to the extrem- 
ities of the curve, is called the 
central angle (J). AV and BVj 
the two equal tangents from the 
vertex to the PC and PT, are 
called the tangent distances (T). 
The chord AB is called the long 
■ chord (LC).' The intercept HG 
from .the middle of the long 
chord to the middle of the arc 
is called the middle ordinate (M). 
^'''- ^' That part of the secant GV from 

the middle of the arc to the vertex is called the external distance 
(E). From the figure it is very easy to derive the following fre- 
quently used relations: 

r = EtanJJ (4) 

LC = 2R sin i J ...«.., (5) 




M=/^ vers ^J 
^=i^exsecii 



(6) 
(7) 



22. Relation between T, E, and A, Join A and G in Fig. 9. 
The angle VAG = \d, since it is measured by one half of the 
arc AG between the secant and tangent. AGO =90° — ^J. 

AV :VG :: sin AGV : sin VAG; 
sin AGF =sin AGO =cos iJ ; 

T :E : : cos i J :siniJ; 

T=E cot iJ 



(8) 



The same relation may be obtained by dividing Eq. 4 by Eq. 
7, since tan a~exsec a = cot ^a. 

23. Elements of a 1° curve. From Eqs. 1 to 8 it is seen that 
the elements of a curve vary directly as R. It is also seen to 
be very nearly true that R varies inversely as D. If the ele- 
ments of a 1° curve for vaiious central angles are calculated and 
tabulated, the elements of a curve of 7)° curvature may be 
approximately found by dividing by D the corresponding ele- 
ments of a 1° curve having the same central angle. For small 



§24. 



ALIGNMENT. 



23 



central angles and low degrees of curvature the, errors involved 
by the approximation are insignificant, and even for larger 
angles the errors are so small that for mam/ purposes they may be 
disregarded 

In Table II is given the value of the tangent distances, 
external distances, and long chords for a 1° curve for various 
j central angles The student should famih'arize himself with the 
degree of approximation involved by solving a large number of 
cases under various conditions by the exact and by the approxi- 
mate methods, in order that he may know when the approxi- 
mate method is sufficiently exact for the intended purpose. 
The approximate method also gives a ready check on the 
exact method. 

24. Exercises, (a) What is the tangent distance of a 4° 20' 
curve having a central angle of 18° 24'? 

(b) Given a 3° 30' curve and a central angle of 16° 20', how 
far Avill the curve pass from the vertex? [Use Eq, 7] 

(c) An 18° curve is to be laid off using 25 -foot (nominal) 
chord lengths. What is the true length of the subchords? 

(d) Given two tangents making a central angle of 15° 24'. 
It is desired to connect these tangents by a curve which shall 
pass 16.2 feet from their intersection. How far down the 
tangent will the curve begin and what will be its radius? (Use 
Eq. 8 and then use Eq 4 inverted.) 

25. Curve location by deflections. The angle between a 
secant and a tangent (or between two secants intersecting on an 
arc) is measured b\' one half of the intercepted arc. Beginning 
at the PC (A in Fig. 10), if the 
first chord is to be a full chord 
we may deflect an angle VAa v 
(=iD\ and the point a, which is 
100 feet from .4, is a point on the 
curve.. For the next station, b, 
deflect an additional angle bAa 
(=W) and, with one end of the 
tape at a, swing the other end 
until the 1004oot point is on the 
line Ab. The point b is then on a 
the curv^e. If the final chord cB 
is a subchord, its additwnal deflec- Fig. 10. 

tion (id) is something less than JD The last deflection (BAV) is 




24 RAILROAD CONSTRUCTION. § 26. 

of course iJ. It is particularly important, when a curve begins 
or ends with a subchord and the d(?flections are odd quantities, 
that the last additional d<iflection should be carefully com- 
puted and added to the previous deflection, to check the mathe- 
matical work by the agreement of this last computed deflec- 
tion with ^J. 

Example. Given a 3° 24' curve having a central angle of 
18° 22' and beginning at sta. 47 + 32, to compute the deflec- 
tions. The nominal length of curv-e is IS* 22' -f- 3° 24' = 18.367 -i- 
3.40 = 5.402 stations or 540.2 feet. The curve therefore ends 
at sta. 52 + 72.2. The deflection for sta. 48 is ^VoXi(3°24') 
=0.68 Xl°.7==l°- 156 = 1° 09' nearly. For each additional 100 
feet it is 1° 42' additional. The final additional deflection for 
the final subchord of 72.2 feet is 

^XK3° 24') =1°.2274 = 1° 14' nearly. 

The deflections are 

P. C . . . Sta. 47 + 32 0° 

48 0° +1°09' = 1°09' 

49 1° 09' + 1° 42' =2° 51' 

50. . 2° 51' + 1° 42'=4° 33' 

51 4° 33' + 1° 42' = 6° 15' 

52 6° 1 5' + 1° 42' = 7° 57' 

•P. T 52 + 72.2 7°57' + l°14'=9°ll' 

As a check 9° ll' = K18° 22') = JJ. (See the Form of Notes 
in § 17.) 

26. Instrumental work. It is generally impracticable to 
locate more than 500 to 600 feet of a curve from one station. 
Obstructions will sometimes require that the transit be moved up 
every 200 or 300 feet. There are two methods of setting off 
the anglers when the transit has been moved up from the PC, 

(a) The transit may be sighted at the previous transit station 
with a reading on the plates equal to the deflection angle from 
that station to the station occupied, but with the angle set off on 
the other side of 0°, so that when the telescope is turned to 0° it 
will sight along the tangent at the station occupied. Plunging 
the telescope, the forward stations may be set off by deflecting 
the proper deflections from the tangent at the station occupied 



26. 



ALIGNMENT. 



25 



•This is a very common method and, when the degree of curva- 
ture is an even number of degrees and when the transit is only 
set at even stations, there is but little ol)jection to it. But the 
degree of curvature is sometimes an odd quantity, and the exi- 
gencies of difficult location frequently require that substations 
be occupied as transit stations. Method (a) will then require 
the recalculation of all deflections for each new station occupied. 
The mathematical work is largely increased and the probability 
of error is very greatly increased and not so easily detected. 
Method (h) is just as simple as method (a) even for the most 
simple cases, and for the more difficult cases just referred to the 
superiority is very great. 

(b) Calculate the deflection for each station and substation 
throughout the curve as though the whole curve were to be lo- 
cated from the PC. The computations 
may thus be completed and checked (as 
above) before beginning the instrumental 
work. If it unexpectedly becomes neces- 
sary to introduce a substation at any 
point, its deflection from the PC may be 
readily interpolated. The stations actually 
set from the PC are located as usual. 
Rule. When the transit is set on any 
forward station, backsight to any previous 
station with the plates set at the deflection 
angle for the station sighted at. Plunge 
the telescope and sight at any forward 
station with the deflection angle originally 
computed for that station. When the 
plates read the deflection angle for the 
station occupied, the telescope is sighting 
along the tangent at that station — which 
is the method of getting the forward tan- 
gent when occupying the PT. Even though 
the station occupied is an unexpected sub- 
station, when the instrument is properly 
oriented at that station, the angle reading 
for any station, forward or back, is that originally computed 
for it from the PC. In difficult work, where there are ob- 
structions, a valuable check on the accuracy may be found by 
sighting backward at any visible station and noting whether 




Fig. 11. 



20 



RAILROAD CONSTRUCTION. 



§26. 



its dofleetion agrees with that originally computed. As a 
numerical illustration, assume a 4° curve, with 28° curvature, 
with stations 0, 2, 4, and 7 occupied. After setting stations 
1 and 2, set up the transit at sta. 2 and backsight to sta. 
with the deflection for sta. 0, which is 0°. The reading on sta. 
1 is 2°; when the reading is 4° the telescope is tangent to 
the curve, and when sighting at 3 and 4 the deflections will be 
6° and 8°. Occupy 4; sight to 2 with a reading of 4°. When 
the reading is 8° the telescope is tangent to the curve and, by 
plunging the telescope, 5, 6, and 7 may be located with the 
originally computed deflections of 10°, 12°, and 14°. When oc- 
cupying 7 a backsight may be taken to any visible station with 
the plates reading the deflection for that station; then w^hen 




-W',. 


^ 






— 


-^^=r- — - 




— 


— 




\ 


•^"" .^ 


\ 





^ 





l' 


\ — , 


1. 


\ 


1 
/ 






Fig. 12. 



Fig. 13. 



the plates read 14° the telescope will point along the forward 
tangent. 

The location of curves by deflection angles is the normal 
method. A few other methods, to be described, should be con- 
sidered as exceptional. 



§27. 



ALKiXMEXT. 



27 



27. Curve location by two transits. A curve might be located 
more or less on a swamp where accurate chaining Avould be 
exceedingly difficult if not impossible. The long chord AB 
(Fig. 12) may be determined by triangulation or otherwise, 
and the elements of the curve computed, including (possibly) 
subchords at each end. The deflection from A and B to each 
point may be computed. A rodman may then be sent (by 
whatever means) to locate long stakes at points determined 
by the simultaneous sightings of the two transits". 

28. Curve location by tangential offsets. When a curve is 
very flat and no transit is at hand the following method may be 
used (see Fig. 13) : Produce the back tangent as far forward as 
necessary. Compute the ordinates Oa^, Ob', Oc', etc., and the 
abscissae a'a, h'h, c'c, etc. If Oa is a full station (100 feet), then 



Oa'^Oa' =100 cos iZ), also = /^ sin D; 

Oh'=Oa' \^a'h' =100 cos JD + lOO cos %D, 

also = R sin 2Z> ; 
Oc' = Oa' + a'y + b'c' = 100(cos W + cos |D -h cos §/)) , 

also=i^ sin SD: 



etc. 



(9) 



a'a = 100 sin JZ), also = R vers D ; 

h'b=a'a + y'b =100 sin JD4-IOO sin ID, 

also = /e vers 2Z); y (10) 
c'c=yb + c''c =100(sin ^D + sin |i) 4-sin fD), 

also =7? vers SD; 



etc. 

The functions JD, |D, etc., may be more conveniently used 
luithout logarithms, by adding the several natural trigonometrical 
functions and pointing off two decimal places. It may also l)e 
noted that Ob' (for example) is one half of the long chord for 
four stations; also that b'b is the middle ordinate for four 
stations. If the engineer is provided with tables giving the long 
chords and middle ordinates for various degrees of curvature, 
these quantities may be taken (perhaps by interpolation) from 
such tables. 

If the curve begins or ends at a substation, the angles and 
terms will be correspondingly altered. The modifications may 



28 RAILROAD CONSTRUCTION". § 29. 

be readily deduced on. the same principles as above, and should 
be worked out as an exercise by the student. 

In Table II are given the long chords for a 1° curve for various 
values of J. Dividing the value as given by the degree of the 
curve, we have an approxinnate value which is amply close for 
low degrees of curvature, especially for laying out curves withr^ 
out a transit. For example, given a 4° 30' curve, required the 
ordinate Oc\ This is evidently one half of a chord of six stations, 
with i=27°. Dividing 2675.1 (which is the long chord of a 
1° curve with J =27°) by 4.5 we have 594.47; one half of this is 
the required ordinate, Oc' =297.23. The exact value is 297.31, 
an excess of .08, or less than X)3 of 1%. The true values 
are always slightly in excess of the value as computed from 
Table II. 

Exercise. A 3° 40' curve begins at sta. 18 4-70 and runs to 
sta. 23 + 60. Required the tangential offsets and their corre- 
sponding ordi nates. The first ordinate = 30 cos KtVo >< 3° 40') = 
30 X. 99995 =29.9985; the offset = 30 sin 0° 33'=30X .0096 = 
0.288. For the second full station (sta. 20) the ordinate = 
i long chord for i =2(1° 06' f 3° 40') with /)=3°40'. Divid- 
ing 476.12, from Table II, by 3|, we have 129.85. Otherwise, 
by Eq. 9, the ordinate = 30 X cos 0° 33' + 100 cos (1° 06' + 1° 50') 
= 30.00 + 99.87 = 129.87. The offset for sta. 20 = 30 sin 0° 33' + 
100 sin (1° 06' + 1° 50') = 0.238 + 5.12 = 5.41. Work out 
similarly the ordinates and offsets for sta. 21, 22, 23, and 
23 + 60. 

29. Curve location by middle ordinates. Take first the sim- 
pler case when the curve begins at an even station. ' If we con- 
sider (in Fig. 14) the curve produced back to z, the chord za = 
2X100 c6s iD, A'a = 100 cos JD, and A'A =am = 2^ = 100 sin JZ). 
Set off AA^ perpendicular to the tangent and A 'a parallel to 
the tangent. .4A' =aa' =66' =cc', etc. = 100 sin JT). Set off 
aa^ perpendicular to a' A. Produce Aa^ until a'6=A'a, thus 
determining h. Succeeding points of the curve may thus be 
determined indefinitely. 

Suppose the curve begins with a subchord. As before 
ra = A m' =c' cos 4<i', and rA = aw' =c' sin Jd'. Also sz = An^ = 
c"cosifr, and sA=zri'=c'' sinid'', in which (c/' + c/") =Z>. 
The points z and a being determined on the ground, «a' may 
be computed and set off as before and the curve continued in 



39. 



ALIGNMENT. 



29 



full stations. A subchord at the end of the curve may be located 
by a similar process. 

30. Curve location by offsets from the long chord. (Fig. 16.) 
Consider at once the general case in which the curve commences 
with a subchord (curvature, d'), continues with one or more full 




Fig. 14. 



Fig. 15. 



Fig. 16. 



chords (curvature of each. Z)), and ends with a subchord with 
curvature d". The numerical work consists in computing first 
AB, then the various abscissae and ordinates. AB=2R sin J J. 



Ah'=Aa'+a'b' =</ cos K^ -^') + 100 cos K^ -2rf'-D); 

Ac =Aa' ■ha'b'' + b'c' = c' cos ^(J -d')-{-100 cos U J -2d' - D) 

■^100 cos U^- 2d'' -D); 
also 

==AB-B(/ ^2R sin ^J -c" cos K^ -d"). 

a'a = a'a =c' sin ^(J— c?'); 

h'h^a'a^-mh^rf sin i(^ -rf') + 100 sin \{A-2d' -B)\ 

c^c =h'h~nb==cf s\Ti^{d-d')-\-ims\n^{J~2d'-D) 

-100 sin ^{ A -2d" -D) 
also =c" sin ^{A^d"). 



(11) 



!^ (12) 



'J 



The above formula? are considerably simplified when the 



30 



ItAlLROAD CONSTKUCTION. 



§31 



curve begins and ends at even stations. When the curve is 
very long a regular law becomes very apparent in the formation 
of all terms between the first and last. There are too few terms 
in the above equations to show the law. 

31. Use and value of the above methods. The chief value 
of the above methods lies in the possibility of doing the work 
without a transit. The same principles are sometimes em- 
ployed, even when a transit is used, when obstacles prevent the 
use of the normal method (see § 32, c). If the terminal tan- 
gents have already been accurately determined, these methods 
are useful to locate points of the curve when rigid accuracy is 
not essential. Track foremen frequently use such methods to 
lay out unimportant sidings, especially when the engineer and 
his transit are not at hand. Location by tangential offsets (or 
by offsets from the long chord) is to be preferred when the 
curve is flat (i.e., has a small central angle J) and there is no 
obstruction along the tangent, or long chord. Location by 
middle ordinates may be employed regardless of the length of 
the curve, and in cases when both the tangents and the long 
chord are obstructed. The above methods are but samples 
of a large number of similar methods which have been devised. 
The choice of the particular method to be adopted must be 
determined by the local conditions. 

32. Obstacles to location. In this section will be given only 
a few of the principles involved in this 
class of problems, with illustrations. The 
engineer must decide, in each case, which 
is the best method to use. It is frequently 
advisable to devise a special solution for 
some particular case. 

a. When the vertex is inaccessible. As 
shown in § 26, it is not absolutely essential 
that the vertex of a curve should be 
located on the ground. But it is very evi- 
dent that the angle between the terminal 
tangents is determined will far less prob- 
able error if it is measured by a single 
measurement at the vertex rather than as 
the result of numerous angle measurements 
Fig. 17. along the curve, involving several posi- 

tions of the transit and comparatively short sights Some- 




§ 32. ALIGNMENT. 31 

times the location of the tangents is already determined on 
the ground (as b}^ bn and am, Fig. 17), and it is required to 
join the tangents by a curve of given radius. Method. Measure 
ab and the angles Vba and baV. A is the sum of these angles. 
The distances bV and aV are computable from the above data. 
Given J and R, the tangent distances are computable, and then 
Bb and a A are found by subtracting bV and aV from the tan- 
gent distances. The curve may then be run from A, and the 
work may be checked by noting whether the curve as run endc 
at B — previously located from b. 

Example. Assume a?>=546 82; angle a = 15° 18'; angle 
b = 18° 22' ; D =3° 40' ; required a A and bB, 
J = 15° 18' + 18° 22' = 33° 40' 

Eq. (4) R (3° 40') 3. 19392 

tan JJ =tan 16° 50' 9.4808 

r = 472.85 2.67475 

sin 18° 22' ab 2.73784 

^^ ~ sin 33° 40' log sin 18° 22' 9.49844 

co-log sin 33° 40' 0.25621 

aF = 310.81 2.49250 

AF = 472.85 



aA =162.04 

"^ sin 33° 40' log sin 15° 18' 9.42139 

co-log sin 33° 40' 0.25621 

6F = 260.29 2.41545 

^F =472. 85 

bB =212.56 

b. When the point of curve (or point of tangency) is inacces- 
sible. M some distance (As, Fig 18) an unobstructed line pn 
may be run parallel with AV. nv=py=As=R vers a. 

•. vers a=As-~R, 

ns=ps=R sina. 



32 



RAILROAD CONSTRUCTION. 



§33. 



At 2/, which is at a distance ps back from the computed posi- 
tion of A, make an offset sA 
to p. Run pn parallel to the 
tangent. A tangent to the 
curve. at n makes an angle of a 
with np. From n the curve is 
run in as usual 

If the point of tangency is 
obstructed, a similar process, 
somewhat reversed, may be 
used. /? is that portion of J still 
to be laid off when m is reached. 
tm—tl=^R sin l^. mz=tB==lx=R 
vers /?. 

c. When the central part of 

the curve is obstructed, a is the 

central angle between two points 

of the curve between which 

a may equal any angle, but it is prefer- 




FiG. 18. 



a chord may be run. 
able that a should be a multiple 
of D, the degree of curve, and that 
the points m and n should be on 
even stations. mn=2R sm^a, A 
point s may be located by an offset 
ks from the chord mn by a similar 
method to that outlined in § 30. 

The device of introducing the 
dotted curve mn having the same 
radius of curvature as the other, 
although neither necessary nor 
advisable in the case shown in 
Fig. 19, is sometimes the best 
method of surveying around an 
obstacle. The offset from any point on the dotted curve to 
the corresponding point on the true curve is twice the ^' ordinate 
to the long chord,'' as computed in § 30. 

33. Modifications of location. The following methods may 
be used in allowing for the discrepancies between the ''paper 
location" based on a more or Jess rough preliminary survey and 
the more accurate instrumental location.' (See § 15.) They are 




Fig. 19. 



33, 



ALIGNMENT. 



33 



also frequently used in locating new parallel tracks and modify- 
ing old tracks. 

a. To move the forward tangent parallel to itself a distance x, 
the point of curve (a) remaining fixed. (Fig. 20.) 



VV 






X 



sin h V F' sin A' 

The triangle BmB^ is isosceles and Bm^^B'm, 

B'r rr' 



E'-B=.0'0=^7nB = 



vers B'mB vers A' 



(13) 



R'=R-\- 



X' 

vers J' 



(14) 



The solution is very similar in case the tangent is moved in- 
ward to Y"B" , Note that this method necessarily changes the 




Fig. 20. 




Fig. 21. 



radius. If the radius is not to be changed, the point of curve 
must be altered as follows: 

b. To move the forward tangent parallel to itself a distance a;, 
the radius being unchanged, (fig. 21.) In this case the whole 



34 



RAILROAD CONSTRUCTION 



§34. 



curve is moved bodily a distance 00' =A A' ==VV' =BB' , and 
moved parallel to the first tangent AV 

B'n X 



BB' = 



=AA', 



(15) 



sin nBB' sin J 

c. To change the direction of the forward tangent at the point 
of tangency. (Fig- 22.) This problem involves a change (a) in 

the central angle and also requires a 
new radius. An error in the deter- 
mination of the central angle fur- 
nishes an occasion for its use. 

Ry J J a, AVj and BV are known. 




a. 



Bs = R vers A 
/. R'=R 



Bs=R' vers A' . 
vers A 



Fig. 22. 



As 



vers (A — a) 
RsinA. A's=R' sin A' 



(16) 



AA'=A's-As=R' sin A'-R sin A. 



(17) 



The above solutions are given to illustrate a large class of 
problems which are constantly arising. All of the ordinary 
problems can be solved by the application of elementary geome- 
try and trigonometry. 

34. Limitations in location. It may be required to run a 
curve that shall join two given tangents and also pass through a 
given point The point (P, Fig. 
23) is assumed to be deter- 
mined by its distance (VP) 
from the vertex and by the 
angle AVP^/^. 

It is required to determine 
the radius (R) and the tangent 
distance (AV). A is known. 

PFG = K180°-J)-^ 
=90o_(ij + /?). 

PP' =2VP sin PVG 

=2FPcos(iJ+/?). 
PSV^^hA. 




_..psin/? 



SP = VP 



Fig. 23. 



sin ^A 



§ 35. ALIGNMENT. 35 

.4 ^ = VSP~X~SP' = VSP{SP-^PP') 

\ sin J J [_ sm JJ ^- . /J 



_ sin^;9^ 2 sin /9 cos (jJ + S) 

~ Xsin^i-/"^ sin* J 

SlFx J J 

AV==AS + SV 

= i^JsinCiJ +^) +\/sin2^ + 2sin^5siniJcos(iJ +^9)]. (18) 
sin2-j 

R=AV cot U. 

In the special case in which P is on the median Hne OF, 
^ = 90°- J J, and (JJ+,5)=90°. Eq. 18 then reduces to 

^F = -^il +COS iJ)=VP cot ii, 
sm iJ 

as might have been immediately derived from Eq. 8. 

In case the point P is given by the offset PK and by the 
distance VK, the triangle PKV may be readily solved, giving the 
distance VP and the angle ^, and the remainder of the solution 
will be as abo\e. 

35. Determination of the curvature of existing track, (a) Using 
a transit. Set up the transit at any point in the center of the 
track. Measure in each direction 100 feet to points also in the 
center of the track. Sight on one point with the plates at 0°. 
Plunge the telescope and sight at the other point. The angle 
between the chords equals the degree of curvature. 

(b) Using a tape and string. Stretch a string (sav 50 feet 
long) between two points on the inside of the head of the outer 
rail. Measure the ordinate (x) between the middle of the string 
and the head of the rail. Then 

„ chord^ , , . ,^^^ 

it= — (very nearh') (19) 

gX 

For, in Fig. 24, since the triangles AOE and ADC are similar, 




36 RAILROAD CONSTRtJGTION. § 36. 

AO : AE :: AD : DC or R^^AD^-^x. When, as is usual, 
the arc is very short compared with the 
radius, AD = ^AB^ very nearly. Making 
this substitution we have Eq. 19. With a 
chord of 50 feet and a 10° curve, the result- 
ing difference in x is .0025 of an inch— far 
within the possible accuracy of such a 
method. The above method gives the 
radius of the inner head of the outer rail. 
It should be diminished by Jg for the radius 
of the center of the track. With easy curvature, however, this 
will not affect the result by more than one or tw^o tenths of one 
per cent. 

The inversion of this formula gives the required middle or- 
dinate for a rail on a given curve. For example, the middle 
ordinate of a 30-foot rail, bent for a 6° curve, is 

a: =900 --(8X955) =.118 foot = 1.4 inches. 

Another much used rule is to require the foreman to have a 
string, knotted at the center, of such length that the middle 
ordinate, measured in inches, equals the degree of curve. To 
find that length, substitute (in Eq. 19) 5730 ^D for R and 
Z)-^ 12 for X. Solving for chord, we obtain chord = 61.8 feet. 
The rule is not theoretically exact, but, considering the uncertain 
stretching of the string, the error is insignificant. In fact, the 
distance usually given is 62 feet, which is close enough for all 
purposes for which such a method should be used. 

36. Problems. A systematic method of setting down the 
solution of a problem simplifies the work. Logarithms should 
always be used, and all the work should be so set down that a 
revision of the work to find a supposed error may be readily 
done. The value of such systematic work will become more 
apparent as the problems become more complicated. The two 
solutions given below will illustrate such work. 

a. Given a 3° curve beginning at Sta. 27 + 60 and running 
to Sta. 32 + 45. Compute the ordinates and offsets used in 
locating the curve by tangential offsets. 

b. With the same data as above, compute the distances to 
locate the curve by offsets from the long chord. 

c. Assume that in Fig. 17 ah is measured as 217.6 feet, the 



§ 3Q. AXJ<^NMENT. 37 

angle a6F = 17°42', and the angle baV=2rU\ Join ihe 
tangents by a 4° 30' curve. Determine bB and aA. 

d. Assume that in a case similar to Fig. 18 \i was noted 
that a distance (As) equal to 12 feet would clear the building. 
Assume that J =38° 20' and that Z)=4°40'. jRequired the 
value of a and the position of n. Solution: 

wers a=As-^R ^s = 12 log = 1.07918 

R (for 4° 40' curve) log = 3 08923 

a=8°0V log vers a = 7.98994 

ns=R sin a log sin a == 9 . 14445 

log ^^3.08923 
ng = 171.27 log = 2 . 23369 

e. Assume that the forward tangent of a 3° 20' eurve having 
a central angle of 16° 50' must be moved 3.62 feet inward, with- 
out altering the P.C. Required the change in radius. 

/. Given two tangents making an angle of 36° 18'. It is 
required to pass a curve through a point 93.2 feet from the 
vertex, the line from the vertex to the point making an angle 
of 42° 21' with the tangent. Required the radius and tangent 
distance. Solution: Applying E.q. 18, we have 

2 log= 0.30103 

,/9=42°21' logsin= 9.82844 

^ 4J = 18°09' log sin = 9.49346 

(i J + ^) = 60° 30' log cos = 9.69234 

.20667 9.31527 

log sin2 /9 = 9 .65688 .45382 

2|9. 81987.... .66049 

9.90993 'T81271 

^ nat. sin 60° 30' .8703 

1 .6830 log= 0.226T0 

FP = 93.2 log = 1 96941 

2.19551 

log sin J J = 9.49346 

Tang, dist. A F = 503. 36 log =2.70205 

log cot i J = 10.48437 

E = 1536.1 log= 3.18642 

Z)=3°44' 



38 



RAILROAD CONSTRUCTION. 



§37. 



COMPOUND CURVES. 

37. Nature and use. Compound curves are formed by a 
succession of two or more simple curves of different curvature. 
The curves must have a common tangent at the point of com- 
pound curvature (P.C.C). In mountainous regions there is 
frequently a necessity for compound curves having several 
changes of curvature. Such curves may be located separately 
as a succession of simple curves, but a combination of two 
simple curves has special properties w^hich are w^orth investigat- 
ing and utilizing. In the following demonstrations R2 always 
represents the longer radius and 7?^ the shorter, no matter which 
succeeds the other. T^ is the tangent adjacent to the curve of 
shorter radius (R^, and is invariably the shorter tangent. A^ is 
the central angle of the curve of radius R^, but it may be greater 
or less than Jg 

38. Mutual relations of the parts of a compound curve having 
two branches. In Fig. 25, AC and CB are the two branches of 




Fig. 25. 



the compound curve having radii of R^ and R2 and central angles 
of Ji and ^2- Produce the arc AC to n so that AO{n = J. The 
chord Cn produced must intersect B. The line ns, parallel to 
CO2, will intersect BO^ so that Bs=sn==0P^ = R2-R^. Draw 
Am perpendicular to 0{n. It will be parallel to hk. 



§ 38. ALIGNMENT. 39 

Br = sn vers Bsn, = ( /?2 — Ri) vers J 2 ; 
m7i = A0j^ vers AO^n =7?^ vers J; 
Ak^A V sin A F/c = T, sin J ; 
Ak^hm^ mn -\-nh= mn -\- Br. 
.-. T, sin J=Ri vers J + (i?2 -^^1) vers J 2. . . (20) 
Similarly it may be shown that 

T2 sin J =i?2 vers J -(R^-Ri) vers J^. . . (21) 

The mutual relations of the elements of compound curves 
may be solved by these two equations. For example, assume 
the tangents as fixed (i therefore known) and that a curve of 
given radius R^ shall start from a given point at a distance T^ 
from the vertex, and that the curve shall continue through a 
given angle J^. Required the other parts of the curve. From 
Eq. 20 we have 

Ti sin J —Ri vers J 



R2 — Ri = 



vers J 



,. R^^X + '^-~!''^^ (22) 

^ ^ vers(J — ii) ^ ^ 

T2 may then be obtained from Eq. 21. 

As another problem, given the location of the two tangents, 
with the two tangent distances (thereby locating the PC and 
PT), and the central angle of each curve; required the two 
radii. Solving Eq. 20 for R^, we have 



Ri 



T^ sin J —R2 vers J 2 



vers J — vers J 2 
Similarly from Eq. 21 we may derive 

r, sin J — i?2(vers J — vers Jj) 

Ki = -^ -. . 

vers J I 

Equating these, reducing, and solving for R2, we have 

Tj sin J vers J1 — T2 sin J (vers J — vers J 2) /r)Qx 

^ vers J2 vers J^— ^vers J— vers Ji)(vers J— vers J2) ' 

Although the various elements may be chosen as above with 

considerable freedom, there are limitations. For example, in 

Eq. 22, since 7?2 is always greater than R^, the term to be 

added to R^ must be essentially positive — i.e., T^ sin J must be 

vers i/ ^ 

greater than R^ vers J. This means that T^>R^ a ^ ^^' ^ ^^ 



40 



RAILROAD CONSTRUCTION. 



§39. 



!ri>7?i tan ^J, or that Ti is greater than the corresponding 
tangent on a simple curve. Similarly it may be shown that Tg 
is less than R2 tan |J or less than the corresponding tangent 
on a simple curve. Nevertheless T2 is always greater than T^. 
In the limiting case when J?2=^it ^2 = ^1? ^^^ ^2 = ^1- 

39. Modifications of location. Some of these modifications 
may be solved by the methods used for simple curves. For 
example : 

a. It is desired to move the tangent VB, Fig. 20, parallel to 
itself to V^B\ Run a new curve from the P.C.C. wliich shall 
reach the new tangent at B', where the chord of the old curve 





Fig. 26. 



Fig. 27. 



intersects the new tangent. The solution is almost identical 
with that in § 33, a. 

b. Assume that it is desired to change the forward tangent 
(as above) but to retain the same radius. In Fig. 27 

(7?2~^i) cos J2 =02n'y 

(R2-R,) cosi2' =02V. 

X = 02n — 02n' = {R2 — Ri) (cos J 2 — <^os J2O • 



cos J/ = cos J- 



X 



(24) 



R2—R1 

The P.C.C. is moved backward along the sharper curve an 
angular distance of J2' — ^2 = ^1 — ^/. 

In case the tangent is moved inward rather than outward, 
the solution will apply by transposing J2 and Jg'- Then we 
shall have 

X 



cos Ji^ = COS ^2 + 



R2—RI 



(25) 



§39. 



ALIGNMENT. 



41 



The P.C.C. is then moved forward. 

c. Assume the same case as (b) except that the larger radius 
comes first and that the tangent adjacent to the smaller radius 
is moved. In Fig. 28 

(Ro—R{) cos J, =0{n; 
(R,-R,) cos J/ =0,'n\ 

x=0^'n^ —0{ii 
= (7^2— ^i)(cos i/— cos ii). 



cos i/ = cos ii 



R2 — -^1 



(26) 




The P.C.C. is moved forward 
along the easier curve an angular 
distance of i/ — Ji = i2~^2'' Fig. 28. 

In case the tangent is moved inward j transpose as before and 
we have 



cos J/ = cos J I 



X 



(27) 



R2—R1 

The P.C.C. is moved backward. 

d. Assume that the radius of one curve is to be altered with- 
out changing either tangent. Assume conditions as in Fig. 29. 

For the diagrammatic solution 
assume that i?2 is to be increased 
by 02^,. Then, since R-/ must 
pass through 0^ and extend be- 
yond Oi a distance O^S, the 
locus of the new center must lie 
on the arc drawn about 0^ as 
center and with OS as radius. 
The locus of O2' is also gJA-en 
by a line O2P parallel to BV 
and at a distance of R2 (equal 
to S . . . P.C.C.) from it. The 
new" center is therefore at the 
intersection O2 . An arc with ra- 
dius R2 will therefore be tangent 
at B' and tangent to the old 
Draw O^n' perpendicular to O2B, 




FjG. 29. 



curve 'produced at new P.C.C. 



42 RAILROAD CONSTRUCTION. § 40. 

With O2 as center draw the arc OiW, and with O2 as center draw 
the arc O^m^ viB=m'B' =R^. 

.'. mn=m'n' = (R2 —Ri) vers J2'=(i?2 — ^1) ^'^rs J2- 

.-. versJ/ = ^^fc||^^ versJ, (28) 

0{n = {R2 — R)) sin A2', 
0,n' = (R2'-R,) sin J/. 
BB' = 0{n' - 0,n = {R2' - R,) sin J 2' ~ (^2 - ^1) sin J 2. (29 

This problem may be further modified by assuming that the 
radius of the curve is decreased rather than increased, or that 
the smaller radius follows the larger. The solution is similar 
and is suggested as a profitable exercise. 

It might also be assumed that, instead of making a given 
change in the radius Ro, a given change BB^ is to be made, ij' 
and R2 are required. Eliminate R2 from Eqs. 28 and 29 
and solve the resulting equation for ^2- Then determine 7?2' 
by a suitable inversion of either Eq. 28 or 29. 

As in §§32 and 33, the above problems are but a few, although 
perhaps the most common, of the problems the engineer may 
meet with in coiTipound curves. All of the ordinary problems 
may be solved by these and similar methods. 

40. Problems, a. Assume that the two tangents of a com- 
pound curve are to be 348 feet and 624 feet, and that J^ =22° 16' 
and J2 =28° 20'. Required the radii. 

[.4ns. 7^1 = 326.92; 7^2 = 1574.85.] 

h. A line crosses a valley by a compound curve which is first 
a 6° curve for 46° 30' and then a 9° 30' curve for 84° 16'. It is 
afterward decided that the last tangent should be 6 feet farther 
up the hill. What are the required changes? [Note. The 
second tangent is evidently moved outward. The solution cor- 
responds to that in the first part of § 39, c. The P.C.C. is 
moved forward 16.39 feet. If it is desired to know how far the 
P.T. is moved in the direction of the tangent (i.e., the projection 
of BB\ Fig. 28, on T^'^'), it may be found by observing that it 
is equal to nn' = (7?2 — 7?i)(sin Jj —sin J/). In this case it equals 
0.65 foot, which is very small because J^ is nearly 90°. The 
value of i^ (46° 30') is not used, since the solution is independent 
of the value of Jj. The student should learn to recognize 



r 



§41. 



AUGXMEXT. 



43 



which quantities are mutually related and therefore essential 
to a solution, and which are independent and non-essential.J 



TRANSITION CURVES. 



41. Superelevation of the outer rail on curves. When a mass 
is moved in a circular path it requires a centripetal force to keep 
it moving in that path. By the principles of mechanics we 
know that this force equals Gv^-^gR, in w^hich G is the weight, 
r the velocity in feet per second, g the acceleration of gravity in 
feet per second in a second, and R the radius of curvature. 
If the two rails of a curved track were laid on a level (trans- 
versely), this centripetal force could only be furnished by the 
pressure of the W'heel-flanges against the rails. As this is very 
objectionable, the outer rail is elevated so that the reaction of 
the rails against the wheels shall 
contain a horizontal component 
equal to the required centripetal 
force. In Fig. 30, if ob represents 
the reaction, oc will represent the 
weight G, and ao will represent the 
required centripetal force. From 
similar triangles we may write 
sn : sm :: ao : oc. Call g = 32.17. 
Call i? =5730-1), which is suffi- 
ciently accurate for this purpose (see 
§ 19). Call r=o2S0r-3600, in which T 




Fig. 30. 



IS 



the velocity in miles 
per hour, mn is the distance between rail centers, which, for 
an 80-lb. rail and standard gauge, is 4.916 feet sm is slightly 
less than this. As an average value we may call it 4.900, which 
is its exact value when the superelevation is 4J inches. Calling 
sn^e, we have 

Gv^l 4.9X52802^22) 



e=sm— =4.9 ^ ^ 
oc gR G 



32.17X36002X5730 



e = .0000b72V''D. 



(30) 



It should be noticed that, according to this formula, the re- 
quired superelevation varies as the square of the velocity, which 
means that a change of velocity of only 10% would call for a 
change of superelevation of 21%. Since the velocities of trains 
over any road are extremely variable, it is impossible to adopt 



44 



RAILROAD COJSTSTRUCTIOX. 



§42. 



any superelevation which will fit all velocities even approx- 
imately. The above fact also shows why any over-iefinement 
in the calculations is useless and why the above approximations, 
which are really small, are amply justifiable. For example, the 
above formula contains the approximation that R = 57S0^D. 
In the extreme case of a 10° curve the error involved would be 
about 1%. A change of about J of 1% in the velocity, or say 
from 40 to 40.2 miles per hour, would mean as much. The error 
in e due to the assumed constant value of sm is never more than 
a very small fraction of 1%. The rail-laying is not done closer 
than this . The following tabular form is based on Eq. (30) : 



SUPERELEVATION OF THE OUTER RAIL (iN FEET) FOR VARIOUS 
VELOCITIES AND DEGREES OF CURVATURE. 



Velocity in 
Miles per 


Degree of Curve. 


Hour. 


1° 


2° 


3° 


4^ 


5° 


6° 


7° 


8° 


9° 


10° 


30 


.05 
.09 
.14 
.20 


.10 
.18 
.29 
.41 


.15 

.27 
■ 43 

r62~ 


.20 
■ 37 

.82 


.26 
■ 46 
. 71 


.31 


.36 


.41 


.46 


1.51 


40 
50 
60 


.86 


.64 


.73 


"sT 





42. Practical rules for superelevation. A much used rule for 
superelevation is to '' elevate one half an inch for each degree of 
curvature.'' The rule is rational in that e in Eq. 30 varies 
directly as D. The above rule therefore agrees with Eq. 30 
when V is about 27 miles per hour. However applicable the 
rule may have been in the days of low velocities, the elevation 
thus computed is too small now. The rule to elevate one inch 
for each degree of curvature is also used and is precisely similar 
in its nature to the above rule. It agrees with Eq. 30 when 
the velocity is about 38 miles per hour, which is more nearly 
the average speed of trains. 

Another (and better) rule is to '' elevate for the speed of the 
fastest trains." This rule is further justified by the fact that a 
four-wheeled truck, having two parallel axles, will always tend 
to run to the outer rail and will require considerable flange pres- 
sure to guide it along the curve. The effect of an excess of super- 
elevation on the slower trains will only be to relieve this flange 
pressure somewhat. This rule is coupled with the limitation 



§42. 



ALIGNMENT. 



45 



that the elevation should never exceed a limit of six inches — 
sometimes eight inches. This limitation imphes that locomo- 
tive engineers must reduce the speed of fast trains around sharp 
curves until the speed does not exceed that for which the actual 
superelevation used is suitable. The heavy line in the tabular 
form (§41) shows the six-inch limitation. 

Some roads furnish their track foremen with a list of the super- 
elevations to be used on each curve in their sections. This 
method has the advantage that each location may be separatelj^ 
studied, and the proper velocity, as affected by local conditions 
(e.g., proximity to a stopping-place for all trains), may be de- 
termined and applied. 

Another method is to allow the foremen to determine the 
superelevation for each curA^e by a simple measurement taken 
at the curve. The rule is developed as follows: By an inversion 
of Eq. 19 we have 

x=chord^-^8R (31) 

Putting X equal to e in Eq. 30 and solving for ''chord,'' we 
have 

chord 2 = m00572V^DFR 

= 2.621 F^ 
chord = imV (32) 

To apply the rule, assume that 50 miles per hour is fixed as 
the velocity from which the superelevation is to be computed. 
Then 1.627 = 1.62X50 = 81 feet, which is the distance given to 
the trackmen. Stretch a tape (or even a string) A\ath a length 
of 81 feet between two points on the inside head of the outer rail 
of the outer head of the inner rail. The ordinate at the middle 
point then equals the superelevation. The values of this chord 
length for varying velocities are given in the accompanying 
tabular form. 



Velocity in miles per hour. . . 20 
Chord length in feet 32 .4 



25 
40.5 



30 
48.6 



35 
56.7 



40 
64. 



45 
872.9 



50 
81.0 



55 
89.1 



60 
97.2 



The following tabular form shows the standard (at one time) 
on the N. Y., N. H. & H. R. R. It should be noted that the 
elevations do not increase proportionately with the radius, and 
that t)^^y »Te higher for descending grades than for level or 



46 



RAILROAD CONSTRUCTION. 



§43. 



ascending grades. This is on the basis that the velocity on curves 
and on ascending grades will be less than on descending grades. 
For example, the superelevation for a 0° 30' curve on a de- 
scending grade corresponds to a velocity of about 54 miles per 
hour, while for a 4° curve on a level or ascending grade the super- 
elevation corresponds to a velocity of only about 38 miles per 
hour. 



TABLE OF THE SUPERELEVATION OF THE OUTER RAIL ON CURVES. 

H. R. R. 



Degree of 
curve. 


Level or as- 


Descending 


cending grade. 


grade. 




inches. 


inches. 


0° 30' 


Of 


1 


1 00 


H 


If 


1 15 


U 


2 


1 30 


2 


2i 


1 45 


2i 


2i 


2 00 


21 


2f 


2 15 


2| 


3 


2 30 


21 


3i 


2 45 


3 


3| 


3 00 


3i 


3f 


3 15 


31 


3i 


3 30 


3f 


4 


3 45 


Si 


4i 


4 00 


4 


4i 



43. Transition from level to inclined track. On curves the 
track is inclined transversely; on tangents it is level. The tran- 
sition from one condition to the other must be made gradually. 
If there is no transition curve, there must be either inclined 
track on the tangent or insufficiently inclined track on the curve 
or both. Sometimes the full superelevation is continued through 
the total length of the curve and the " run-off '' (having a length 
of 100 to 400 feet) is located entirely on the tangents at each 
end. In other practice it is located partly on the tangent and 
paitly on the curve. Whatever the method, the superelevation 
is correct at only one point of the run-off. At all other points 
it is too great or too small. This (and other causes) produces 
objectionable lurches and resistances when entering and leav- 
ing curves. The object of transition curves is to obviate these 
resistances. 

On the lichigh Valley R. R, the run-off is made in the form 
of a reversed vertical curve, as shown in the accompanying 
figure. According to this system the length of run-off Aariefs 



§44. 



ALIGNMENT. 



47 



from 120 feet, for a superelevation of one inch, to 450 feet, 
for a superelevation of ten inches. Such a superelevation 
as ten inches is ver}^ unusual practice, but is successfully 
operated on that road. The curve is concave upward for two- 
thirds of its length and then reverses so that it is convex upward. 

TABLE FOR RUN-OFF OF ELEVATION OF OUTER RAIL OF CURVES. 
Drop in inches for each 30-foot rail commencing at theoretical point of curve. 



£.2 


V 


Y 


¥ 


¥ 


r 


r 


¥ 


\" 


\¥ 


W 


IF 


\" 


¥ 


¥ 


¥ 


¥ 


¥ 


¥ 


1 

Iti ,t 


^e" 


o 


w" 












































H 


V 




30 


30 






























30 




30 




120 


2" 




30 








30 




















30 


30 






30 




150 


r 




30 








30 














30 




30 




30 






30 




180 


r 




30 




30 






30 












30 




30 


30 




30 




30 


. . . 


'^40 


b" 




30 




30 








30 








30 




30 


30 


30 




30 




30 




270 


&' 




30 




30 






30 




30 




. . . 


30 




30 


30 


30 




30 




30 




300 


T 




30 




30 






30 




30 




30 


30 




30 


30 




30 


30 




30 




330 


^" 




30 




30 




30 






30 


30 


30 


30 




30 


30 




30 




30 




30 


360 


^" 


30 






30 




30 




30 


30 




30 


30 


30 


30 


30 


30 


30 




30 




30 


420 


10'' 


30 




30 




30 


■ • 




30 


30 


30 


30 


30 


30 


30 


30 


30 


30 




30 




30 


450 




The figure (and also the lower line of the tabulated form) 
shows the drop for each thirty-foot rail length. For shorter 
lengths of run-ofi, the drop for each 30 feet is shown by the cor- 
responding hues in the tabular form. Note in each horizontal 
hne that the sum of the drops, under which 30 is found, equals 
the total superelevation as found in the first column. For 
example, for 4 inches superelevation, length of curve 240 feet, 
the successive drops are \" , ¥\ i'\ |'', f", §'', i", and i'' 
whose sum is 4 inches. Possibly the more convenient form 
would be to indicate for each 30-foot point the actual super- 
elevation of the outer rail, which would be for the above case 
(running from the tangent to the curve) J", |", |", U", 2|", 
3r, 3r', 4-. 

44. Fundamental principle of transition curves. If a curve 



48 RAILROAD CONSTRUCTION. § 45. 

has variable curvature, beginning at the tangent with a curve 
of infinite radius, and the curvature gradually sharpens until it 
equals the curvature of the required simple curve and there 
becomes tangent to it, the superelevation of such a transition 
curve may begin at zero at the tangent, gradually increase to 
the required superelevation for the simple curve, and yet have 
at every point the superelevation required by the curvature at 
that point. Since in Eq. (30) e is directly proportional to D, 
the required curve must be one in which the degree of curve 
increases directly as the distance along the curve. The mathe- 
matical development of such a curve is quite complicated. It 
has, however, been developed, and tables have been computed for 
its use, by Prof. C. L. Crandall. The following method has the 
advantage of great simplicit}^, while its agreement with the true 
transition curve is as close as need be, as will be shown. 

45. Multiform compound curves. If the transition curve com- 
mences with a very flat curve and at regular even chord lengths 
compounds into a curve of sharper curvature until the desired 
curvature is reached, the increase in curvature at each chord 
point being uniform, it is plain that such a curve is a close ap- 
proximation to the true spiral, especially since the rails as laid 
will gradually change their curvature rather than maintain a 
uniform curvature throughout each chord length and then 
abruptly change the curvature at the chord points. Such a 
curve, as actually laid^ will be a much closer approximation 
to the true curve than the multiform compound curve by which 
it is set out. There will actually be a gradual increase in curva- 
ture which increases directly as the length of the curve. 

46. Required length of spiral. The required length of spiral 
evidently depends on the amount of superelevation to be gained, 
and also depends somewhat on the speed. If the spiral is laid 
off in 25-foot chord lengths, with the first chord subtending a 1° 
curve, the second a 2° curve, etc., the fifth chord will subtend 
a 5° curve, and the increase from this last chord to a 6° curve 
is the same as the uniform increase of curvature between the 
chords. The same spiral extended would run on to a 12° curve 
in (12 — 1)25=275 feet. The last chord of a spiral should have 
a smaller degree of curvature than the simple curve to which it 
is joined. If the curves are very sharp, such as are used in street 
work and even in suburban trolley work, an increase in degree 
of curvature of 1° per 25 feet will not be sufficiently rapid, as 



§47. 



ALIGNMENT. 



49 



such a rate would require too long curves, 2°, 10°, or even 20° 
increase per 25 feet may be necessary, but then the chords 
should be reduced to 5 feet. Such 
a rapid rate of increase is justified 
by the necessary reduction in 
speed. On the other hand, very 
high speed will make a lower rate 
of increase desirable, and there- 
fore a spiral whose degree of cur- 
vature increases only 0° 30' per 25 
feet may be used. Such a spiral 
would require a length of 375 feet 
to run on to an 8° curve, which is 
inconveniently long, but it might 
be used to run on to a 4° curve, 
where its length would be only 175 
feet. Three spirals have been de- 
veloped in Table IV, each with 
chords of 25 feet, the rate of in- 
crease in the degree of curvature 
being 0° 30', 1° and 2° per chord. 
One of these will be suitable for 
any curvature found on ordinary 
steam-railroads . 

47. To find the ordinates of a 
i°-per-25-feet spiral. Since the 
first chord subtends a 1° curve, its central angle is 0° 15' and the 
angle aQV (Fig. 31) is 7' 30". The tangent at a makes an 
angle of 15' with VQ. The angle between the chord ba and the 
tangent at a is i(30')=15', and the angle 6a?)" = J(30') + 15' 
= 30'. Similarly 

the angle chc'' = 4(45') + 30' + 15' = 67' 30" = 1° 07' 30", 
and the angle dcd'' = 2° 0'. 

The ordinate aa' =25 sin 7' 30", and 
Qa'=25 cos 7' 30". 
Qy=Qa' + a'b' 

= 25 (cos 7' 30" + cos 30'). 
66'=6'6" + 66" 

= 25 (sin 7' 30" + sin 30') „ 
Similarly the ordinates of c, d, etc., may be obtained. 




Fig. 31. 



50 



RAILROAD CONSTRUCTION. 



§48. 



48. To find the deflections from any point of the spiral. 
aQV = 7'S0'\ Tan hQV =h¥ --- Qh' \ tan cQV =cc' -.Qc'; etc. 
Thus we are enabled to find the deflection angles from the tan- 
gent at Q to any point of the spiral. 

The tangent to the curve at c (Fig. 32) makes an angle of 




Fig. 32. 
1°30' with QV, or cmF = l°30'. Qcm = cmV -cQm. The 
value of cQm is known from previous work. The deflection 
from c to Q then becomes known. 

ac7n = cmV — cap = cmV — caq — qap. caq is the deflection an- 
gle to c from the tangent at a and will have been previously 
computed numerically, gap = 15'. acm therefore becomes 
known. 

6cm = iof 45'=22'30"; 

dcn = i of GO' =30'. 



§49. 



ALIGNMENT. 



51 



ecn=ecd" —ncd'^, ncd^'=^cmV, tan ecd" = {ee' —d"d')-^c'e'j all 
of which are known from the previous work 

By this method the deflections from the tangent at any point 




Fig. 33. 



of the curve to any other point are determinable. These values 
are compiled in Table IV The corresponding values of these 
angles when the increase in the degree of curvature per chord 
length is 30', and when it is 2°, are also given in Table IV. 

49. Connection of spiral with circular curve and with tangent. 
See Fig. 33.* Let AV and BV be the tangents to be connected 



* The student should at once appreciate the fact of the necessary distor- 
tion of the figure. The distance MM' in Fig. 33 is perhaps 100 time^ its real 
proportional value. 



52 RAILROAD CONSTRUCTION. § 49. 

by a D° curve, having a suitable spiral at each end. If no 
spirals were to be used, the problem would be solved as in simple 
curves giving the curve AMB Introducing the spiral has the 
effect of throwing the curve away from the vertex a distance 
MM' and reducing the central angle of the D° curve by 20. 
Continuing the curve beyond Z and Z' to .4' and B', we will 
have AA^ = BB' =^M M\ ZK = the x ordinate and is therefore 
known. Call MM' =m. A'N =x—R vers 0. Then 



m-=MM'=AA' = z-: = TT-^- ..... (33) 

cos JJ cos JJ 



NA = A A' sin lJ = (x — R vers 0) tan JJ. 

VQ=QK-KN {NA+AV 

=^y~R sin ^{x — R vers 0) tan JJ-f^ tan §J 

= 2/— /^ sin + a: tan JJ + i? cos tan Jz/. . . , . (34) 

When A'N has already been computed, it may be more con- 
venient to write 



yQ=2/4 i?(tan JJ-sin 0)+AWtaniJ (35) 

VM' = VM+MM' 

T^ -, . ^ ^ vers 6 .^^. 

=i^exseciJ + o HT (36) 

cos § J COS J J 



AQ==VQ-AV 

=y — R sin (j>"\-(x — R vers ^) tan h J (37) 



Example. To join two tangents making an angle of 34° 20' 
bv a 5° 40' curve and suitable spirals. Use l°-per-25-feet spirals 
wUh five chords. Then = 3M5', a; = 2.999, iJ = 17°10', and 
2/ = 124.942. 



§50. 



ALIGNMENT. 



53 



[Eq. 33J 



[Eq. 36] 



[Eq. 351 



[Eq. 37] 





R 


3.00497 




vers (/) 


7 33063 


2.166 




0.33560 


x= 2,999 






A'N= 0.833 




9.92064 




cos J J 


9.98021 


m=MM'=AA' = 0.872 


i^ 


9 9404:3 




3.00497 




exsec i J 


8.6686 5 


FM=47.164 




1.67360 


m= 0.872 






FM' = 18.036 






y = 124:. 942 na t . ianU -- 


= .30891 




nat. sin ^ = 


=.06540 






.24351 


9.38651 




R 


3.00497 


246.314 


A'N 


2.39148 


[See above] 


9.92064 




tan Ji 


9.48984 


' 0.257 


AN 


9.41048 


rQ =371 .513 






R 


3.00497 




tan iJ 


9.48984 


312 471 


AV 


2.49481 



AQ= 59.042 

50. Field-work. When the spiral is designed during the 
original location, the tangent distance VQ should be computed 
and the point Q located. It is hardly necessar}^ to locate all of 
the points of the spiral until the track is to be laid. The ex- 
tremities should be located, and as there will usually be one 
and perhaps two full station points on the spiral, these should 
also be located. Z may be located by setting off QK =y and 
KZ=Xj or else by the tabular deflection for Z from Q and the 
distance ZQ, which is the long chord. Setting up the instrument 
at Z and sighting back at Q with the proper deflection, the tan- 
gent at Z may be found and the circular curve located as usual, 
its central angle being A—2(j). A similar operation will locate 
Q' from Z\ 

To locate points on the spiral. Set up at Q, with the plates 



54 



RAILROAD CONSTRUCTION. 



§50. 



reading 0° when the telescope sights along VQ. Set off from 
Q the deflections given in Table IV for the instrument at Q, 
using a chord length of 25 feet, the process being like the method 
for simple curves except that the deflections are irregular. If 
a full station-point occurs within the spiral, interpolate between 
the deflections for the adjacent spiral-points. For example, 
a spiral begins at Sta. 56 + 15. Sta. 57 comes 10 feet beyond 
the third spiral point. The deflection for the third point is 
35' 0''; for the fourth it is 56' 15''. ^ of the difference 
(21' 15") is 8' 30"; the deflection for Sta. 57 is therefore 43' 30". 
This method is not theoretically accurate, but the error is small. 
Arriving at Z, the forward ahgnment may be obtained by sight- 
ing back at Q (or at any other point) with the given deflection 




for that point from the station occupied. Then when the plates 
read 0° the telescope will be tangent to the spiral and to the 
succeeding curve. All rear points should be checked from Z, 
If it is necessary to occupy an intermediate station, use the de- 
flections given for that station, orienting as just explained for Z, 



§ 51. ALIGNMENT. 55 

checking the back points and locating all forward points up to Z 
if possible. 

After the center curve has been located and Z' is reached, the 
other spiral must be located but in reverse order, i.e., the sharp 
cu. vature of the spiral is at Z' *and the curvature decreases 
toward Q\ 

51. To replace a simple curve by a curve with spirals. This 
may be done by the method of § 49, but it invohes shifting the 
whole track a distance m, which in the given example equals 
0.87 foot. Besides this the track is appreciably shortened, 
which would require rail-cutting But the track may be kept 
at practically the same length and the lateral deviation from the 
old track may be made very small by slightly sharpening the 
curvature of the old track, moving the new curve so that it is 
wholly or partially outside of the old curve, the remaindei of it 
with the spirals being inside of the old curve. It is found by 
experience that a decrease in radius of from 1% to 5% will 
answer the purpose. The larger the central angle the less the 
change. The solution is as indicated in Fig. 34. 

0'N=R' cos ^-{x. 
0'V=0'N sec i J 

= R' cos (j) sec \A-[-x sec J J. 
m = MM'=MV-M'V 

= Rc^sec\A-{0'V-R') 

= R exsec ^J — R^ cos ^ sec ^J—x sec iJ+R\ . . . (38) 
AQ=QK~KN+NV-VA 

= y — R' sin (j)-]~(R' cos (jy + x) tan ^A—R tan J J 

= ?/— E' sin ^4-J?' cos (/)tan JJ-(7^— .7^ tan JJ. . . (39) 



The length of the old curve from Q to Q' =2.4 Q + 100--. 

The length of the new curve from Q to Q' =2L^-100~^-^ , 

in which L is the length of each spiral- 
Example, Suppose the old curve is a 7° 30' curve with a 
central angle of 38° 40'. As a trial, compute the relative length 
of a new 8° curve with spirals of seven chords, ^ = 7°0'; 
JJ = 19°20'; R (for the 7° 30' curve) =764.489; B' (for the 
8° curve) =716.779; a;==7.628. 



56 RAILROAD CONSTRUCTION. § 52. 

[Eq. 38] R 2.88337 

exsec ^ J 8.77642 

45.687 1.65979 

/2' = 716^779 =z= 

762.466 ft' 2.85538 

cos 9 . 99675 
sec i J 0.02521 

753.953 2.87734 

X 0.88241 

sec ^ J 0- 02521 

8.084 . 90762 

762.037 762.037 

m= 0.429 

[Eq. 39] 2/ = 174.722 ft' 2.85538 

sin<?> 9 . 08589 

87.353 1.94128 

ft' 2.85538 

cos<?> 9.99675 

tan * J 9.54512 

249.606 2 ■ 39725 

ft = 764. 489 

x^ 7.628 

TseTsei 2.87901 

tan iJ Q '^^'^^2 
265.543 2.4241^ 

424.328 352.896 
352 . 896 

^Q = 71.432 

The length of the old curve from Q to Q' is 

100-^ = 100?^ = 515.556 

2^Q = 2 x:71.432 = 142.864 

658.420 

■KT ^nr^A-^'P , ^^ 38.667 - 14.000 _„ ^_^ 
New curve: 100 — jy, — = 100 ^-^r = 308.333 

^2L = 2 X 175 = 350.000 

658.333 658.333 
Difference in length = . 087 

Considering that this difference may be divided among 22 
joints (using 30-foot rails) no rail-cutting would be necessary. 
If the difference is too large, a slight variation in the value of 
the new radius R' will reduce the difference as much as neces- 
sary. A truer comparison of the lengths would be found by 
comparing the lengths of the arcs. 

52. Application of transition curves to compound curves. 
Since compound curves are only employed when the location is 
limited by local conditions, the elements of the compound curve 
should be determined (as in §§38 and 39) regardless of the 



§52. 



ALIGNMENT. 



57 



transition curves, depending on the fact that the lateral shifting 
of the curve when transition curves are introduced is very small. 
If the limitations are very close, an estimated allowance may be 
made for them. 

Methods have been devised for inserting transition curves 
between the branches of a compound curve, but the device is 




Fig. 35. 



complicated and usually needless, since when the train is once 
on a curve the wheels press against the outer rail steadily and 
a change in curvature will not produce a serious jar even though 
the superelevation is temporarily a little more or less than it 
should be. 



58 RAILROAD CONSTRUCTION. § 53. 

If the easier curve of the compound curve is less than 3° or 
4°, there may be no need for a transition curve off from that 
branch. This problem then has two cases according as transition 
curves are used at both ends or at one end only. 

a. With transition curves at both ends. Adopting the method 
of § 49, calling Ji = Ji, we may compute m^=MM^\ Similarly, 
calling J2 = iJ, we may compute m.^=MM2\ But M/ and M/ 
must be made to coincide. This may be done by moving the 
curve Z^M/ and its transition curve parallel to Q^V a distance 
M/M^, and the other curve parallel to QT^ a distance M2^M^, 
In the triangle M/M3M2', the angle at Mi' = 90° — i,, the angle 
at M2'=90° — ^2, and the angle at M^ = J, 

rrn nr f^T 7.>* / tix / ^iu (90° — J9) . . COS io 1 

Then M/M.^M/M/ ^^. — ,— ={'rn.—m^ -. — }. 

^ ^ ^ ^ sm i ^ ^ ^^ sm J 

y (40) 

o- -1 1 nr ,ixr 7i>f- /7i.r /Sin (90° — ii) , . COS Ji I 

Similarly Mo'M.^M/M.' ~ — :; — - = (m. —m.,—. — -- 

b. With a transition curiae on the sharper curve only. Com- 
pute mi=MM/ as before; then move the ciwve Z^M^ parallel 
to Q^V a distance of 

M/M,=mi^^-7 (41) 

The simple curve AIA is moved parallel to VA sl distance of 

,^, , cos Ji /^^x 

MM.==m.-. — f (42) 

sin J 

• 

If Ji and ^2 ^^^ both small, M/M^ and MM^ may be more 
than m,, but the lateral deviation of the new curve from the old 
will always be less than m^. 

53. To replace a compound curve by a curve with spirals. 
The solution is somewhat analogous to that of § 51. Compute 
nil ^^^ ^^^ sharper branch of the curve, placing Ay^ = ^A in Eq. 
38. Since m^ and m^ for the two branches of the curve must 
be identical, a value for Rj must be found which will satisfy 
the determined value of m2=m^^. Solving Eq. 38 for R\ we 
obtain 

cos — COS iJ ^ 



§ 53. ALIGNMENT. 59 

Substituting in this equation the known value of m^ {=m^ 
and caUing R'=R2j R = R2, and Jg^i^? solve for i^g'- Obtain 
the value of AQ for each branch of the curve separately by Eq. 
39, and compare the lengths of the old and new lines. 

Example. Assume a compound curve with Z)i=8°, Z)2=4°, 
J^ = 36°, and J2=32°. Use l°-per-25-feet spirals; ^, = 7°0'; 
^^ = 1°30'. Assume that the sharper curve is sharpened from 
8° 0' to 8° 12'. 
[Eq. 38] 



169 209 . 




exsec 36° 


2.85538 
9.. 37303 
2 22842 


/2i' = 699.326 
868.535 


857.970 . . . 
9 . 429 . . 


cos 01 
sec J I 

sec Ji 




2.84468 
9.99675 
0.09204 

2.93347 




0.88241 
0.09204 

0.97445 


867 . 399 
wi= 1.136 


867.399 







[Eq. 43] R2 3.15615 

' vers 32° 9.18170 

217.700 2 . 33785 

cos 32° 9.92S4.2 

0.963 9.98380 

a;2= 0.763 

1.726 1.726 _____ 

215.974 2.33440 

nat. cos <^ = . 99966 
nat. cos J 2= .84805 

.15161 9.18073 

i22' = 1424.54 [4°1'22"] 3.15367 

[Eq.39] 2/1 = 174.722 ^^, ^^^^^ 

Bin 4>i 9 . 08589 

85.226 1.93057 

/?/ 2.84468 

cos<^i 9.99675 

tan ii [ii == 36°] 9.86126 

504.302 ..... 2.70269 

iei-716.779 

xi= 7.628 

709.151 2.85074 

tan^J 9.86126 

679.024 — 

, 600.461 515.235 2.71200 

^Qi= 78.563 600.461 



60 RAILROAD CONSTRUCTION. § 53. 

[Eq. 39] R/ 3.15367 

2/2= 74.994 sin<^2 8-41792 

37.290 1.57159 



R/ 3.15367 

cos 4>2 9 . 99y85 

tan iJ(J2 = 32°) 9 . 79579 

889.843. . 2.9'*93'l 



j i2>= 1432.69 
X2=^ 0.76 

1431.93 3.15592 
tan iJ 9.79579 

894.770 2.95171 



964.837 932.060 
932 . 060 

AQ2^ 32.777 

For the length of the old track we have 

^1 ...^36 



100^' = 100^= 450. 



D^ 8 



)0 



lOOyr-^ = 100 '-- = 800. 

D2 4° 

AQy = 78,563 

AQo = 32.777 

1361.340 



For the length of the new track we have : 

10oA_li..lOO^= 353.659 

100^^-100|^= 758.140 

Spiral on 8° 12' curve 175.000 
•' 4° 01' 22" " 75. 

Length of new track = 136 1 . 799 

" ♦♦ old " = 1361.340 

Excess in length of new track = . 459 feet. 

Since the new track is slightly longer than the old, it shows 
that the new track runs too far outside the old track at the 
P.C.C. On the other hand the offset m is only 1.136. The 
maximum amount by which the new track comes inside of the 
old track at two points, presumably not far from Z' and Z, is 
very difficult to determine exactly. Since it is desirable that 
the maximum offsets (inside and outside) should be made as 
nearly equal as possible, this feature should not be sacrificed to 
an effort to make the two lines of precisely equal length so that 
the rails need not be cut. Therefore, if it is found that the offsets 
inside the old track are nearly equal to m (1.136), the above 
figures should stand. Otherwise m may be diminished (and the 
above excess in length of track diminished) by increasing R/ 
very slightly and making the necessary consequent changes. 



§ 54. ALIGNMENT. 61 



VERTICAL CURVES. 

54. Necessity for their use. Whenever there is a change in 
the rate of grade, it is necessary to eliminate the angle that 
would be formed at the point of change and to connect the two 
grades by a curve. This is especially necessary at a sag between 
two grades, since the shock caused by abruptly forcing an up- 
ward motion to a rapidly moving heavy train is very severe both 
to the track and to the rolling stock. The necessity for vertical 
curves was even greater in the days when link couplers were in 
universal use and the ^' slack'' in a long train was very great. 
Under such circumstances, when a train was moving down a 
heavy grade the cars would crowd ahead against the engine. 
Reaching the sag, the engine would begin to pull out, rapidly 
taking out the slack. Six inches of slack on each car would 
amount to several feet on a long train, and the resulting jerk on 
the couplers, especially those near the rear of the train, has fre- 
quently resulted in broken couplers or even derailments. A 
vertical curve will practically eliminate this danger if the curve 
is made long enough, but the rapidly increasing adoption of 
close spring couplers and air-brakes, even for freight trains, is 
obviating the necessity for such very long curves. 

55. Required length. Theoretically the length should de- 
pend on the change in the rate of grade and on the length of the 
longest train on the road. A sharp change in the rate of grade 
requires a long curve; a long train requires a long curve; but 
since the longest trains are found on roads with light grades and 
small changes of grade, the required length is thus somewhat 
equalized. It has been claimed that a total curve length equal 
to one-third of the train length for each tenth of a per cent of 
change of rate of grade will certainly prevent the rear of the 
train from crowding against the cars in front, but such a length 
is admittedly excessive. Half of this length is probably ample 
and one-fourth of it is probably safe. Therefore, we may say, 
taking the even fraction ^^ rather than -^^, 

length of vertical curve = (length of longest train) X (change 
of rate of grade in per cent). 

For example, assume a change of rate of grade of 2% ; assume 
that the longest train will be about 720 feet. Then, by the 



62 RAILROAD CONSTRU€TION. § 56. 

above rule, the length of curve should be 720X2 = 1440 feet. 
Such rules are seldom if ever applied except in the most approx- 
imate way. On many roads a uniform length of onl}^ 400 feet 
is adopted for all vertical curves. The required length over 
a hump is certainly much less than that through a sag. Added 
length increases the amount of earthwork required both in cuts 
and fills, but the resulting saving in operating expenses will 
always justify a considerable increase. 

56. Form of curve. In Fig. 36 assume that A and C, equi- 






LEVEL LINE ~-^^ B' 



^ f 



Fig. 36. 

distant from B, are the extremities of the vertical curve. Bisect 
AC at e; draw Be and bisect it at h. Bisect AB and BC at k 
and I. The line kl will pass through h. A parabola may be 
drawn with its vertex at h w^hich will be tangent to AB and BC 
at A and C It may readily be shown * from the properties of 
a parabola that if an ordinate be drawn at any point (as at n) 
we will have 

sn : eh (or hB) : : Am^ \ Ae} 

or sn=eh (44) 

Ae^ 

In Fig. 36 the grades are necessarily exaggerated enormously. 
With the proportions found in practice we may assume that 
ordinates (such as mf, eB, etc.) are perpendicular to either 
grade, as may suit our convenience, without any appreciable 
error. In the numerical case given below, the variation of 
these ordinates from the vertical is 0° 07', while the effect of 
this variation on the calculations in this case (as in the most 
extreme cases) is absolutely inappreciable. It may easily be 
shown that the angle CAB=h.8iU the algebraic difference of the 
rates of grade. Call the difference, expressed in per cent of 
grade, r; then CAB = ^r. Let Z = length (in ^' stations'^ of 100 
feet) of the line AC, which is practically equal to the horizontal 

* See note at foot of p. 63. 



§ 57. ALIGNMENT. 63 

measurement. Since the angle CAB is one-half the total change 
of grade at B, it follows that Be = \lX \r Therefore 

Bh = llr (44a) 

Since Bh ^^or eh) are constant for any one curve, the correction 
sn Bit any point (see Eq. 44) equals a constant times Am"^. 

57. Numerical example. Assume that B is located at St a. 
16 + 20; that the curve is to be 1200 feet long; that the grade 
of AB is -0.87o, and of BC -Vl.2%] also that the elevation 
of B above the datum plane is 162.6. Then the algebraic dif- 
ference of the grades, r, =1.2-(-0 8) =2.0; Z = 12. Bh = llr 
= J- X 12X2 =3.0. A is at Sta. 10 + 20 and its elevation is 
162.6 + (6X0.8) =167.4; C is at Sta 22 + 20 and its elevation is 
162.6 + (6X1.2) =169.8. The elevation of Sta. 11 is found by 
adding sn to the elevation of s on the straight grade line. The 

constant (e/i-v-Ae") equals in this case 3. 04-600^ = 120^0 o- 
Therefore the curve elevations are 

A, Sta. 10 + 20, 162.6 + (6.00X0.8) =167.40 

11 167. 4-( .80X0.8) +120V5TT 802=166.81 

12 167.4-(1.80X0.8) + lo^Wa 1802=166.23 

13 167. 4 -(2. 80X0. 8) +-i2o'ooo 2802= 165.81 

14 167.4-(3.80X0.8) +120^00 3802= 165.56 

15 167.4-(4.80X0.8) +i5o'oo5 4802= 165.48 

16 167. 4 -(5. 80X0. 8) + 12 oW 580?= 165.56 

B, 16 + 20, 162 . 6 + 3 . = 165 . 60 

17 169. 8 -(5. 20 XI. 2) +t^(jIoo 5202= 165.81 

18 169.8-^(4.20X1.2) +i2t>Voo 4202= 166.23 

19 169.8-(3.20X1.2) +^2o'oo(j 3202= 166.81 

20 169.8-(2.20X1.2) +120V00 2202=167.56 

21 169.8-(1.20X1.2) +t2Co 50 1202= 168.48 

22 169. 8-( .20X1.2) +i2oW 202 = 169.56 

C, 22 + 20, 162. 6 + (6. 00X1. 2) =169.80 



DEMONSTRATION OF EQ. 44. 

The general equation of a parabola passing through the point n (Fig. 36) 
may be written 

2/2 + 2/„2 = 2v{x + 2:,,) 5 

•j/2 y jj 
from which x„ =— h -r — — x, 

'* 2p 2p ^ - 

When X = xj y = yj^ and we have 



64 RAILROAD CONSTRUCTION. § 57, 



The general equation of 


a tangent passing 


through the point A 


may 


be 


written 






yijA = 


■ P(x + x^y 








from which 






X = 


vva 








When X = x^, y 


= 1/s[ = 


= 2/, 


1, and 


we have 
VnyA 










sn = 


^n 


. 8n = 


yA^ + yn^- 

2p 

(yA-ynY 

2p 
yA' -Ae' 
^A eh 

A 2 

—rAm 

eh -„. 

Ae^ 


■^ynVA 

Am^ 
2p' 







This proves the general proposition that if secants are drawn parallel to 
the axis of x, intersecting a parabola and a tangent to it, the intercepts be- 
tween the tangent and the parabola are pioportional to the square of the 
distances (measured parallel to y) from the tangent point. 



CHAPTER III. 

EARTHWORK. 

FORM OF EXCAVATIONS AND EMBANKMENTS. 

58. Usual form of cross-section in cut or fill. The normal 
form of cross-section in cut is as shown in Fig. 37, in which 
e . . . gr represents the natural surface of the ground, no matter 




how irregular; ah represents the position and width of the re- 
quired roadbed; ac and hcl represent the ^'side slopes" which 
begin at a and h and which intersect the natural surface at such 




Fig. 38. 






points (c and d) as will be determined by the required slope 
angle iff). 

The normal section in fill is as shown in Fig. 38. The points 
c and d are likewise determined by the intersection of the re- 

65 



66 



RAILROAD CONSTRUCTION 



§59. 



quired side slopes with the natural surface. In case the required 
roadbed (ab in Fig. 39) intersects the natural surface, both cut 




Fig. 39. 

and fill are required, and the points c and d are determined as 
before. Note that /? and /?' are not necessarily^ equal. Their 
proper values will be discussed later. 

59. Terminal pyramids and wedges. Fig. 40 illustrates the 
general form of cross-sections when there is a transition from 
cut to fill, a , , ,g represents the grade line of the road which 




Fig. 40. 

passes from cut to fill at d. sdt represents the surface },-rofile. 
A cross-section taken at the point where either side of the road- 
bed first cuts the surface (the point m in this case) will usually 
be triangular if the ground is regular. A similar cross-section 
should be taken at o, where the other side of the roadbed cuts 
the surface. In general the earthwork of cut and fill terminates 



§ 60. EARTHWOKK. 67 

in two pyramids. In Fig. 40 the pyramid vertices are at n 
and k, and the bases are Jhm and opq. The roadbed is generally 
wider in cut than in fill, and therefore the section Ihrn and the 
altitude In are generally greater than the section opq and the 
altitude pJx. When the line of intersection of the roadbed and 
natural surface {nodkm) becomes perpendicular to the axis of 
the roadbed (ag) the pyramids become wedges whose bases are 
the nearest convenient cross-sections. 

6o. Slopes, a. Cuttings. The required slopes for cuttings 
vary from perpendicular cuts, which may be used in hard rock 
which will not disintegrate b}^ exposure, to a slope of perhaps 
4 horizontal to 1 vertical in a soft material like quicksand or in 
a cla3'ey soil which flows easily w^hen saturated. For earthy 
materials a slope of 1 : 1 is the maximum allowable, and even 
this should only be used for firm material not easily affected by 
saturation, A slope of IJ horizontal to 1 vertical is a safer 
slope for average earthwork It is a frequent blunder that 
slopes in cuts are made too steep, and it results in excessive work 
in clearing out from the ditches the material that slides down, 
at a much higher cost per yard than it would have cost to take 
it out at first, to say nothing of the danger of accidents from 
possible landslides. 

b. Enibankm.ents. The slopes of an embankment vary from 
1 : 1 to 1.5 : 1. A rock fill will stand at 1 : 1, and if some care 
is taken to form the larger pieces on the outside into a rough 
dry wall, a much steeper slope can be allowed. This method is 
sometimes a necessity in steep side-hill w^ork. Earthwork em- 
bankments generally require a slope of 1} to 1. If made 
steeper at first, it generally results in the edges giving way, re- 
quiring repairs until the ultimate slope is nearly or quite J^ -1. 
The difficulty of incorporating the added material with the old 
embankment and preventing its sliding off frequently makes 
these repairs disproportionately costly. 

6i. Compound sections. When the cut consists partly of 
earth and partly of rock, a compound cross-section must be 
made. If borings have been made so that the contour of the 
rock surface is accurately known, then the true cross-section may 
be determined. The rock and earth should be calculated sepa- 
rately, and this will require an accurate knowledge of where the 
rock ''runs out'' — a difficult matter when it must be deter- 



68 RAILROAD CONSTRUCTION. § 62. 

mined by boring. During construction the center part of the 
earth cut would be taken out first and the cut widened until a 
sufficient width of rock surface had been exposed so that the 
rock cut would have its proper width and side slopes. Then the 
earth slopes could be cut down at the proper angle. A^'berm" 
of about three feet is usually left on the edges of the rock cut as 




Fig. 41. 

a margin of safety against a possible sliding of the earth slopes. 
After the work is done, the amount of excavation that has been 
made is readily computable, but accurate preliminary estimates 
are difficult. The area of the cross-section of earth in the figure 
must be determined by a method similar to that developed for 
borrow-pits (see § 89). 

62. Width of roadbed. Owing to the large and often dis- 
proportionate addition to volume of cut or fill caused by the 
addition of even one foot to the width of roadbed, there is a 
natural tendency to reduce the width until embankments become 
unsafe and cuts are too narrow for proper drainage. The cost 
of maintenance of roadbed is so largely dependent on the drain- 
age of the roadbed that there is true economy in making an 
ample allowance for it. The practice of some of the leading 
railroads of the country in this respect is given in the following 
table, in which are also given some data belonging more properly 
to the subject of superstructure. 

It may be noted from the table that the average width 
for an earthwork cut, single track, is about 24.7 feet, with a 
minimum of 19 feet 2 inches. The widths of fills, single track, 
average over 18 feet, with numerous minimums of 16 feet. 
The widths for double track may be found by adding the distance 
between track centers, which is usually 13 feet. 



§62, 



EARTHWORK. 



69 



0) CJ 


. 




























ipfe 1 




^ 




^ ^ 


C3-M 


^COOO COCOCO COC'ICCCO 




a -t-r 


t- c 


1— It— IrH tHt-ii—I r-(T-HT— Ii— 1 (M 




.iLi a)H o I 




I— 1 




n« 


" 












1—1 1— t T— ( r-( tH — J T— 1 T— < »-H 1— 1 1— 1 1— 1 


tH 1—1 






^* 





•• • 


>> 


i 


S 


LO I0i-Oi0i0»0i0»0«0>0tci0 


; io^ 




•^ 




T— 1 1-H t—l rH T—l r-* I— ( I— 1 r— T—l T—i .— I 


" T-i 1-i 


^ 
P^ 




- 


-^ 


_ 1 








'm 


V 

^ 




1— 1 t— 1 T— 1 I— 1 1—1 T— t T— 1 1— 1 1— ' 


^ 


TS 


jO 




1— It— ( tH__t— 1 T-ii— ' ..1— 


tt) 


^ 


"S 


•■ ■■ioio>ow:>iO "lo "loio^o ■ 


• u:) • 


2? 




o 


i-H H* !-( • tH • • • r^y^ • 1- 


a 






1— ' I— 1 T-l 1— I T— 1 1-H T-l TH 1— i 


T-l 


-2 


















. fj 
















^4 




: ^ 

•o ceo 


QOO(N 


•coc'^ 




2 


, 


S 




•CO^CO 


^ COCO 

CO 

CO 


coco>> 


tH 


3 










CO 


: CO 


o 


c3 














K 


ti 
















[5 






• »HS^ 


sS 


^ ^^ X 




o 
ft 


-(-3 

o 




:xxx 

.(NCOCO 


ooXx 


Xo<y 

CO ^ 
CO CO 


+ 

CO 


bC 

1 












?2 












• rC 


















-M 








! ^ ;j 










... 




"^ 


o 


•CO coooQCo 


CO 






■ 5: 5: 

: f^^cc 


'T3 




S 


(N 


• r-l -^ (M 1-1 ^ ^ 






1—i 


■ bios'" 


'g 


Ji^* 






• (N <^ 










•4- 


















- ^ 




^ *^__ i;: 


iC 




^ 








o 


t^ ^ lO o -^ 


X 

+ 




(N 1 


o - >.C0 
be c:,(M 

2ff+ 






__.^^(N ^ 


CO 




^ C<(M1H 


'S 






1-1 


1— 1 




. ^ 


-— -^ ^ 1-1 


g 
















— •♦-' 
















CJ 








. "^'^ 


CO 

4> 








>3 




















^ 




































^c^ 








1^ 










X 


• 




: 


o o 3 : 










c 












oW C c d 




> 

> 


cS C ' "^ 




1 


* 






So V, , «2 


Xim > bJL . = 


ll 
















r _r:_c! :~ s- o) es o — • o 


0) C 








< 


ooc 


r^ 


i^ 


h:i 


>^ 


t-l 


^ 


:2; 


^ 


flH 


p 





70 RAILROAD CONSTRUCTION. § 63. 

63. Form of subgrade. The stability of the roadbed depends 
largely on preventing the ballast and subsoil from becoming 
saturated with water The ballast must be porous so that it 
w ill not retain water, and the subsoil must be so constructed that 
it will readily drain off the rain-water that soaks through the 
ballast. This is accomplished by giving the subsoil a curved 
form, convex upward^ or a surface made up of two or three 
planes, the two outer planes having a slope of about 1 : 24 
(sometimes more and sometimes less, depending on the soil) 
and the middle plane, if three are used, being level. When a 
circular form is used, a crowning of 6 inches in a total width of 
17 or 18 feet is generally used. Occasionally the subgrade is 
made level^ especially in rock-cuts, but if the subsoil is previously 
compressed by rolling, as required on the N. Y. C & H. R. R. R., 
or if the suVjsoil is drained by tile drains laid underneath the 
ditches, the necessity for slopes is not so great. Rock cuts are 
generally required to be excavated to one foot below subgrade 
and then filled up again to subgrade with the same material, if 
it is suitable. 

64. Ditches. '*The stability of the track depends upon the 
strength and permanence of the roadbed and structures upon 
which it rests; whatever will protect them from damage or pre- 
vent premature decay should be carefully observed. The w^orst 
enemy is water, and the further it can be kept aw^ay from the 
track, or the sooner it can be diverted from it, the better the 
track will be protected. Cold is damaging only by reason of 
the w^ater w^hich it freezes; therefore the first and most impor- 
tant provision for good track is drainage." (Rules of the Road 
Department, Illinois Central R. R.) 

The form of ditch generally prescribed has a flat bottom 12" 
to 24" wide and with sides having a minimum slope, except in 
rock-w^ork, of 1 : 1, more generally 1.5 : 1 and sometimes 2 : 1. 
Sometimes the ditches are made V-shaped, which is objection- 
able unless the slopes are low^ The best form is evidently that 
which will cause the greatest flow for a given slope, and this 
J will evidently be the form in which the 
ratio of area to whetted perimeter is the 
W)7M/////'' largest. The semicircle fulfills this con- 

^ ^„ dition better than any other form, but the 

FiG. 42. "^ 

nearly vertical sides would be difficult to 

maintain. (See Fig. 42.) A ditch, with a flat bottom and such 




§ 65. EARTHWORK. 71 

slopes as the soil requires, which approximates to the circular 
form will therefore be the best. 

When the flow will probably be large and at times rapid it 
will be advisable to pave the ditches with stone, especially if the 
soil is easily washed away. Six-inch tile drains, placed 2' under 
the ditches, are prescribed on some roads. (See Fig. 43.) No 
better method could be devised to insure a dry subsoil. The 
ditches through cuts should be led off at the end of the cut so 
that the adjacent embankment will not be injured. 

Wherever there is danger that the drainage from the land 
above a cut will drain down into the cut, a ditch should be made 
near the edge of the cut to intercept this drainage, and this 
ditch should be continued, and paved if necessary, to a point 
where the outflow will be harmless. Neglect of these simple 
and inexpensive precautions frequently causes the soil to be 
loosened on the shoulders of the slopes during the progress of a 
heavy rain, and results in a landslide which will cost more to 
repair than the ditches which would have prevented it for all 
time. 

Ditches should be formed along the bases of embankments; 
they facilitate the drainage of water from the embankment, 
and may prevent a costly slip and disintegration of the em- 
bankment. 

65. Effect of sodding the slopes, etc. Engineers are unani- 
mously in favor of rounding off the shoulders and toes of em- 
bankments and slopes, sodding the slopes, paving the ditches, 
and providing tile drains for subsurface drainage, all to be put 
in during original construction. (See Fig. 43.) Some of the 
higliest grade specifications call for the removal of the top layer 
of vegetable soil from cuts and from under proposed fills to 
some convenient place, from which it may be afterwards spread 
on the slopes, thus facilitating the formation of sod from grass- 
seed. But while engineers favor these measures and their 
economic value may be readih^ demonstrated, it is generally 
impossible to obtain the authorization of such specifications 
from railroad directors and promoters. The addition to the 
original cost of the roadbed is considerable, but is by no means 
as great as the capitalized value of the extra cost of mainte- 
nance resulting from the usual practice. Fig. 43 is a copy of 



RAILROAD CONSTRUCTION. \ 



§65. 



designs * presented at a convention of the Ameiican Society of 
Civil Engineers by Mr. D. J. Whittemore, Past President of 
the Society and Chief Engineer of the Chi., Mil. & St. Paul 




"^. PROPOSED SECTION OF ROADBED IN EXCAVATION.. 




CUSTOMARY SECTION OF ROADBED ON EMBANKMENT. 



IF 




PROPOSED SECTION OF ROADBED ON EMBANKMENT. 

GRAVEL,^ i L 1 




Fig. 



43. — " Whittemore on Railway Excavation and Embankments " 
Trans. Am. Soc. C. E., Sept. 1894. 



R. R. The "customary sections '' represent what is, with some 
variations of detail, the practice of many railroads. The '^ pro- 



♦Trans. Am. Soc. Civil Eng., Sept. 1894. 



§ 66. EARTHWORK. 73 

posed sections" elicited unanimous approval. They should be 
adopted when not prohibited by financial considerations. 

EARTHWORK SURVEYS. 

66. Relation of actual volume to the numerical result. It 

should be realized at the outset that the accuracy of the result 
of computations of the volume of any giA^en mass of earthwork 
has but little relation to the accuracy of the mere numerical 
work. The process of obtaining the volume consists of two 
distinct parts. In the first place it is assumed that the volume 
of the earthwork may be represented by a more or less com- 
plicated geometrical form, and then, secondly, the volume of 
such a geometrical form is computed. A desire for simplicity 
(or a frank willingness to accept approximate results) will often 
cause the cross-section rnen to assume that the volume may be 
represented by a very simple geometrical form which is really 
only a very rough approximation to the true volume. In such 
a case, it is only a waste of time to compute the volume with 
minute numerical accuracy. One of the first lessons to be 
learned is that economy of time and effort requires that the 
accuracy of the numerical work should be kept proportional to 
the accuracy of the cross-sectioning work, and also that the 
accuracy of both should be proportional to the use to be made 
of the results. The subject is discussed further in § 94. 

67. Prismoids. To compute the volume of earthwork, it is 
necessary to assume that it has some geometric form whose vol- 
ume is readily determinable. The general method is to consider 
the volume as consisting of a series of prismoids, Avhich are 
solids having parallel plane ends and bounded by surfaces which 
may be formed by lines moving continuously along the edges of 
the bases These surfaces may also be considered as the sur- 
faces generated by lines moving along the edges joining the cor- 
responding points of the bases, these edges being the directrices, 
and the lines being always parallel to either base, which is a 
plane director. The surfaces thus developed may or may not 
be planes. The volume of such a prismoid is readily determin- 
able (as explained in § 70 et seq.), while its definition is so very 
general that it may be applied to very rough ground. The 
'Hwo plane ends" are sections perpendicular to the axis of the 
road. The roadbed and side slopes (also plane) form three of 



74 



RAILROAD CONSTRUCTION. 



§68. 



the side surfaces. The only approximation lies in the degree of 
accuracy with which the plane (or warped) surfaces coincide with 
the actual surface of the ground between these two sections. 
This accuracy will depend (a) on the number of points which 
are taken in each cross-section and the accuracy with which the 
lines joining these points coincide with the actual cross-sections; 
(b) on the skill shown in selecting places for the cross-sections so 
that the warped surfaces shall coincide as nearly as possible with 
the surface of the ground. In fairly smooth country, cross- 
sections every 100 feet, placed at the even stations, are suf- 
ficiently accurate, and such a method simplifies the computations 
greatly; but in rough country cross-sections must be inter- 
polated as the surface demands. As wnll be explained later, 
carelessness or lack of judgment in cross-sectioning will introduce 
errors of such magnitude that all refinements in the computa- 
tions are utterl}^ wasted. 

68. Cross-sectioning. The process of cross-sectioning con- 
sists in determining at any place the intersection by a vertical 
plane of the prism of earth lying between the roadbed, the side 
slopes, and the natural surface. The intersection with the road- 




FiG. 44. 



bed and side slopes gives three straight lines. The intersection 
with the natural surface is in general an irregular line. On 
smooth regular ground or when approximate results are accept- 
able this line is assumed to be straight. According to the irreg- 



§ 69. EARTHWORK. 75 

ularity of the ground and the accurac}- desired more and more 
''intermediate points" are taken. 

The distance (d in Fig. 44) of the roadbed below (or above) 
the natural surface at the center is known or determined from 
the profile or by the computed establishment of the grade line. 
The distances out from the center of all " breaks " are deter- 
mined with a tape. To determine the elevations for a cut, set 
up a level at any convenient point so that the line of sight is 
higher than any point of the cross- section, and take a rod read- 
ing on the center point. This rod reading added to d gives the 
height cf the instrument (H. I.) above the roadbed. Sub- 
tracting from H. I. the rod reading at any ''break" gives the 
height of that point above the roadbed (hj, ki, hr, etc.). This 
is true for all cases in excavation. For fill, the rod reading at 
center minus d equals the H. I., which may be positive or nega- 
tive. When negative, add to the " H. I." the rod readings of 
the intermediate points to get their depths below "grade"; 
when positive, subtract the " H. I." from the rod readings. 

The heights or depths of these intermediate points above or 
below grade need only be taken to the nearest tenth of a foot, 
and the distances out from the center will frequently be suffi- 
ciently exact when taken to the nearest foot. The roughness of 
the surface of farming land or woodland generally renders use- 
less any attempt to compute the volume with any greater accu- 
racy than these figures would implv unless the form of the ridges 
and hollows is especiall}^ well defined. The position of the slope- 
stake points is considered in the next section. Additional dis- 
cussion regarding cross-sectioning is found in § 82. 

69. Position of slope-stakes. The slope-stakes are set at the 
intersection of the required side slopes with the natural surface, 
which depends on the center cut or fill (d). The distance of 
the slope-stake from the center for the lower side is x = ^h 
+ s(d-\-y); for the up-hill side it is x^ =^b-]-s(d—y^). s is the 
"slope ratio" for the side slopes, the ratio of horizontal to ver 
tical. In the above equation both x and y are unknown. There- 
fore some position must be found by trial which will satisfy the 
equation. As a preliminary, the value of x for the point a = hh 
■i-sd, which is the value of x for level cross-sections. In the 
case of fills on sloping ground the value of x on the down-hill 
side is greater than this; on the xip-hill side it is less. The differ- 
ence in distance is s times the difference of elevation. Take a 



76 



RAILROAD CONSTRUCTION. 



§69. 



numerical case corresponding with Fig. 45. The rod reading 
on c is 2.9; cZ=4.2; therefoie the telescope is 4.2—2.9 = 1.3 
helow grade. &-=1.5 : 1, 6 = 16. Hence for the point a (or for 
level ground) a: = {rX 10 + 1.5X4. 2 = 14. 3. At a distance out 
of 14.3 the ground is seen to be about 3 feet lower, which will 
not only require 1.5X3=4.5 more, but enough additional dis- 
tance so that the added distance shall be 1.5 times the additional 
drop. As a first trial the rod ma}^ be held at 24 feet out and a 
reading of, say, 8.3 is obtained. 8.3 + 1.3=9.6, the depth of 
the point below grade. The point on the slope line (n) which 
has this depth below grade is at a distance from the center 



T 











\ 1 


4 

1 


"T- 


— -jT^- — li" 


i^^ 


f-1— 


^^^ifc^^T 






^^IIZI"!:!!:""!"..!!!! 





^^f^x'<X 



y 



Fig. 45. 



x = 8 + 1.5X9.6 =22.4. The point on the surface (.s) having 
that depth is 24 feet out. Therefore the true point (m) is 
nearer the center. A second trial at 20.5 feet out gives a rod 
reading of, say, 7.1 or a depth of 8.4 below grade. This corre- 
sponds to a distance out of 20.6. Since the natural soil (espe- 
cially in farming lands or woods) is generally so rough that a 
difference of elevation of a tenth or so may be readily found by 
slightly A^arying the location of the rod (even though the dis- 
tance from the center is the same), it is useless to attempt too 
much refinement, and so in a case like the above the combina- 
tion of 8.4 below grade and 20.6 out from center may be taken 
to indicate the proper position of the slope-stake. This is 
usually indicated in the form of a fraction, the distance out being 
the denominator and the height above (or below) grade being 
the numerator; the fact oC cut or ^11 may be indicated by C or F* 
Ordinarily a second trial will be sufficient to determine with 
sufficient accuracy the true position of the slope- stake. Ex- 
perienced men will frequently estimate the required distance 



§ 69. EARTHWORK. 77 

out to within a few tenths at the first trial. The left-hand pages 
of the note-book should have the station number, surface eleva- 
tion, grade elevation, center cut or fill, and rate of grade. The 
right-hand pages should be divided in the center and show the 
distances out and heights above grade of all points, as is ilhis- 
trated in § 84. The notes should read up the page, so that when 
looking ahead along the line the figures are in their proper 
relative position. The ^'fractions" farthest from the center 
line represent> the slope-stake points. 

69a. Setting slope-stakes by means of " automatic " slope- 
stake rods. The equipment consists of a specially graduated 
tape and a specially constructed rod. The tape may readily be 
prepared by marking on the back side of an ordinary 50-foot 
tape which is graduated to feet and tenths. Mark ''0" at " ^h'^ 
from the tape-ring. The same tape may be used for several 
values of " ^h'' by placing the zero at the maximum distance ^h 
from the ring. Then graduate from the zero backward, at true 
scale, to the ring. When J6 is less than this maximum, the 
tape will not be used clear to the ring. In general, the tape 
must be so held that the zero is always ^h from the center stake. 
Mark off ''feet" and ''tenths" on a scale proportionate to the 
slope ratio. For example, with the usual slope ratio of 1.5 : 1 
each "foot" would measure 18 inches and each "tenth" in 
proportion. 

The' rod, 10 feet long, is shod at each end and has an endless 
tape passing within the shoes at each end and over pulleys — to 
reduce friction. The tape should be graduated in feet and 
tenths, from to 20 feet — the and 20 coinciding. By moving 
the tape so that is at the bottom of the rod — or (practically) 
so that the 1-foot mark on the tape is one foot above the bottom 
of the shoe, an index mark may be placed on the back of the 
rod (say at 15 — on the tape) and this readily indicates w^hen the 
tape is "set at zero." 

The method of use may best be explained from the figure and 
from the explicit rules as stated. The proof is given for two 
assumed positions of the level. 

(1) Set up the level so that it is higher than the "center" 
and (if possible) higher than both slope-stakes, but not more 
than a rod-length higher. On very steep ground this may be 
impossible and each slope-stake must be set by separate positions 
of the level. 



78 



RAILROAD CONSTRUCTION. 



§69. 



(2) Set the rod-tape at zero (i.e., so that the 15-foot mark 
on the back is at the index mark) . 

(3) Hold the rod at the center-stake- (^) and note the read- 
ing (ni or n.-). Consider n to be always plus; consider d to be 
plus for cut and minus for fill. 

(4) i^mse the tape on the /ace side of the rod (71 + d). Applied 
literally (and algebraically), when the level is helow the roadbed 
(only possible for fill) , (n + d) = (n2 + {—df)) =712 —df. This being 
numerically negative, the tape is lowered {df—n^. With level 
at ( 1) , for fill, {n-\'d)= (n^ -\-(—df))= (n^ —dj) ; this being positive, 
the tape is raised. With level at (1), for cut, the tape is raised 
(ni + dc) . In every case the effect is the same as if the telescope 
were set at the elevation of the roadbed. 




Fig. 45a. 

(5) With the special distance-tape, so held that its zero is ^h 
from the center, carry the rod out until the rod reading equals 
the reading indicated by the tape. Since in cut the tape is 
raised (n-\-d), the zero of the rod-tape is always higher than the 
level (unless the rod is held at or below the elevation of the road- 
bed — which is only possible on side-hill work), and the reading 
at either slope-stake is necessarily negative. The reading for 
glope-stakes in fill is always positive. 

(6) Record the rod-tape reading as the numerator of a frac- 
tion and the actual distance out (read directly from the other 
side of the distance-tape) as the denominator of the fraction. 

Proof. Fill. Level at (i). Tape is raised {n^—df). When 
rod is held at C/, the rod reading is +0:, which =rfi — (ni—df). 
But the reading on the back side of the distance-tape is also x. 

Fill. Level ^t (2). Tape is raised (^2 — J/), i.e., it is lowered 
(df—n.^. When rod is held at C/, the rod reading is +.t, which 



§70. 



EARTHWORK. 



79 



similarly = Tf2 — {n2—df) = r/2 + {df—n.^. Distance-tape as be- 
fore. 

Cut Level at (i). Tape is raised (ni + c?c)- When rod is 
held at Cc the rod reading is— 2:, which = rci — (ni + (ic), i.e., 
2 = (n^ + c?c) — Tci. The distance-tape will read z. 

Side-hill work. It is easily demonstrated that the method, 
when followed literally, may be applied to side-hill work, al- 
though there is considerable chance for confusion and error, 
when, as is usual, J6 and the slope ratio are different for cut and 
for fill. 

The method appears complicated at first, but it becomes 
mechanical and a time-saver w^hen thoroughly learned. The 
advantages are especially great when the ground is fairly level 
transversely, but decrease when the difference of elevation 
of the center and the slope-stake is more than the rod length. 
By setting the rod-tape ' ' at zero,'' the rod may always be used 
as an ordinary level rod and the regular method adopted, as in 
§ 69. Many engineers who have thoroughly tested these rods 
are enthusiastic in their praise as a time-saver. 

COMPUTATION OF VOLUME. 

70. Prismoidal formula. Let Fig. 46 represent a triangular 
prismoid. The two triangles forming the ends lie in paraUel 
planes, but since the angles of one triangle are not equal to the 
corresponding angles of the other triangle, at least two of the 
surfaces must be warped. If a section, parallel to the bases, is 




— &j- 



^j^ — f<_ 



Fig. 46. 



made at any point at a distance x from one end, the area of the 
section will evidently be 



80 KAILROAD CONSTRUCTION. § 70. 

The volume of a section of infinitesimal length will be Axdx, 
and the total volume of the prismoid will be * 



= i [bA 



^ + (^2 -&i)/ii 27 + ^1(^2-^1)'^ 






i^ A + 4 (^- . -^2 . -^2~ 1 + i^2/^2 



=|[Ai + 4A.. + A2], (45) 

in which A^, A 2, and Am are the areas respectively of the two 
bases and of the middle section. Note that Am is not the mean 
of Aj and A^j although it does not necessarily differ very greatly 
from it. 

The above proof is absolutely independent of the values, ab- 
solute or relative, of 5i, 62? ^1? o^ ^2- For example, /ij may be 
zero and the second base reduces to a line and the prismoid be- 
comes wedge-shaped; or 62 ^^^ ^2 may both vanish, the second 
base becoming a point and the prismoid reduces to a pyramid. 
Since every prismoid (as defined in § 67) may be reduced to a 
combination of triangular prismoids, wedges, and pyramids, and 

* Students unfamiliar with the Integral Calculus may take for granted the 

fundamental formulae that / dx = x, that / xdx = ^x^, and that 5 x'^dx = ^x^\ 

also that in integrating between the T.nits of I and OCzero), the value of the 
integral may be found by simply suV)stituting I for x after integration. 



§ 71. EARTHWORK. 81 

since the formula is true for any one of them individually, it is 
true for all collectively ; therefore it may be stated that * 

The volume of a prismoid equals one sixth of the 'perpendicular 
distance between the bases multiplied by the sum of the areas of 
the two bases plus four times the area of the middle section. 

While it is always possible to compute the volume of any 
prismoid by the above method, it becomes an extremely compli- 
cated and tedious operation to compute the true value of the 
middle section if the end sections are complicated in form. It 
therefore becomes a simpler operation to compute volumes by 
approximate formulae and apply, if necessary, a correction. 
The most common methods are as follows : 

71. Averaging end areas. The volume of the triangular 
prismoid (Fig. 46), computed by averaging end areas, is 

I 

— [ J61/11 + i&2^2]- Subtracting this from the true volume (as 

given in the equation above Eq. 45), we obtain the correction 

^[{b,-h)(h,-h,)] (46) 

This shows that if either the /t's or 6's are equal, the correc- 
tion vanishes; it also shows that if the bases are roughly similar 
and b varies roughly with h (which usually occurs, as will be 
seen later) , the correction will be negative, which means that the 
method of averaging end areas usually gives too large results. 

72. Middle areas. Sometimes the middle area is computed 
and the volume is assumed to be equal to the length tim,es the 

middle area. This will equal — X ^ ^ X ^ ^ ^ . Subtracting 

this from the true volume, w^e obtain the correction 

^^\-h,){K-h,) , (47) 

As before, the form of the correction shows that if either 
the /I's or 6's are equal, the correction vanishes; also under the 
usual conditions, as before, the correction is positive and only 
one-half as large as by averaging end areas. Ordinarily the 
labor involved in the above method is no less than that of 
applying the exact prismoidal formula. 

* The student should note that the derivation of equation (45) does not 
complete the proof, but that the statements in the following paragraph are 
logically necessary for a general proof. 



82 



KAILROAD CONSTRUCTION. 



§73. 



73. Two-level ground. When approximate computations of 
earthwork are sufficiently exact the field-work may be materi- 
ally reduced by observing simply the center cut (or fill) and the 
natural slope a, measured with a clinometer. The area of such 

a section (see Fig. 48) equals 

ah 



But 

from which 

Similarly, 
Substituting, 



i{a + d){xi-\-Xr) — 2" 

Xi tan p = a-\-d-\-xi tan a, 
a-{-d 



xi 



Xf 



tan/?— tan a 
a-\-d 



Area = (a -\- dy 



tan /? + tan a ' 
tan /? 



ah 
~2 



(48) 



tan^ ^ — tan^ a 

The values a, tan /?, tan^ ^/? are constant for all sections, so 
that it requires but little work to find the area of any section. 




Fig. 47. 

As this method of cross-sectioning implies considerable approxi- 
mation, it is generally a useless refinement to attempt to com- 




FiG. 48. 



§ 74. EARTHWORK. 83 

pute the voiume with any greater accurac}^ than that obtained 
by averaging end areas. It may be noted that it may be easily 
proved that the correction to be appHed is of the same form as 
that foimd in § 71 and equals 

j^[(a-/ + xr') - {xi" + Xr")] [(.d" + a) -{d' + a)l 

which reduces to 

When d" =d' the correction vanishes. This shows that when 
the center heights are equal there is no correction — regardless 
of the slope. If the slope is uniform throughout, the form of the 
correction is simplified and is invariably negative. Under the 
usual conditions the correction is negative, i.e., the method 
generally gives too large results. 

74. Level sections. AMien the country is very level or when 
only approximate preliminary results are required, it is some- 
times assumed that the cross-sections are level. The method of 
level sections is capable of easy and rapid computation. The 
area may be written as 

(a + d)^s-^ (50) 

! / ■ 

'iiiii^iimiiiiiiiiimim 
"\ I / \ 

Fig. 49. 

1 





This also follows from Eq. 48 w^hen a=0 and tan /? = 



s here represents the " slope ratio," i.e., the ratio of the horizontal 
projection of the slope to the vertical. A table is very readily 
formed gi^'ing the area in square feet of a section of given depth 
and for any given width of roadbed and ratio of side-slopes. 



84 RAILROAD CONSTRUCTION. § 75 

The area may also be readily determined (as illustrated in the 
following example) without the use of such a table: a table of 
squares will facilitate the work Assuming the cross-sections 
at equal distances (=/) apart, the total approximate volume 
for any distance will be 



^^[Ao + 2{A, + A,+ . . .An-:)+Anl . • . (51) 



The prismoidal correction may be directly derived from 
Eq. 46 as ^[2(a + d')s-2{a + d'')s][(a + d'')-(a + d')l ^yh[ch 
reduces to 

-^{d'-d'^y or -^\d'-d'y. . . (52) 



This may also be derived from Eq 49, since a = 0, tan a = 0, 
and tan/? = 2a-^Z) This correction is always negative, showing 
that the method of averaging end areas, w^hen the sections are 
level, always gives too large results The prismoidal correction 
for any one prismoid is therefore a constant times the square 
of a difference. The squares are always positive whether the 
differences are positive or negative. The correction therefore 
becomes 

-1^2(d'~d")^. (53) 



75. Numerical example: level sections. Given the following 
center heights for the same number of consecutive stations 100 
feet apart; width of roadbed 18 feet; slope IJ to 1. 

The products in the fifth column may be obtained very 
readily and with sufficient accuracy by the use of the slide-rule 
described in § 79. The products should be considered as 

{a-\-d){a-{-d)-^ — . In this problem s = li, — = .6667. To apply 

o o 

the rule to the first case above, place 6667 on scale B over 89 
on scale A, then opposite 89 on scale B will be found 118.8 on 
scale A. The position of the decimal point will be evident from 
an approximate mental solution of the problem. 



§ 76, 



EARTHWORK. 



85 



Sta. 


Center 
Height. 


a-\-d 


(a + d)2 


(a-\-dys 


J7 
18 
19 
20 
21 
22 


2.9 

4.7 
6.8 
11.7 
4.2 
1.6 


8.9 
10.7 
12.8 
17.7 
10.2 

7.6 


79.21 
114.49 
163.84 
313.29 
1 04 . 04 

57 . 76 


118.81 
171.741 
245.76 I 
469.93 r 
1 56 . 06 ] 
86.64 



Areas. 



X2 



118.81 
f 343 . 48 
J 491.52 
1 939 . 86 
t312.12 
86.64 



d'^d" 


1 


8 


2 


1 


4 


9 


7 


5 


2 


6 



(d"^d^)a 



3.24 

4.41 

24.01 

56.25 

6.76 



ah 6X18 ^, 



1752.43X100 

2X27 



2292.43 

10X54 = _540^ 

1752.43 

3245 cub, yards = approx. vol. 



Corr. = - .:!:^Z^ X 94 . 67 = - 91 cub. yds. 



12X6X27 
3245-91 =3154 cub. yds. 



94.67 



= exact vclume. 



The above demonstration of the correction to be apph'ed to 
the approximate vohime, found by a^-eraging end areas, is intro- 
duced mainly to give an idea of the amount of that correction. 
Absokitely level sections are practically unknown, and the error 
involved in assuming any given sections as truly level will 
ordinarily be greater than the computed correction. If greater 
accuracy is required, more points should be obtained in the 
cross-sectioning, which will generally show that the sections 
are not truly level. 

76. Equivalent sections. When sections are very irregular 
the following method may be used, especially if great accuracy 
is not required. The sections are plotted to scale and then a 
uniform slope line is obtained by stretching a thread so that the 
undulations are averaged and an equivalent section is obtained. 
The center depth (d) and the slope angle (a) of this line can be 
obtained from the drawing, but it is more convenient to measure 
the distances (xi and Xr) frpm the center. The area may then 
be obtained independent of the center depth as follows: Let 

b 



s = the slope ratio of the side slopes = cot ^ = 



2a 



(See Fig. 50.) 



Then the 



Area 



^1 /xi-\-Xr\ 

2\ s "J 



(xi-\-Xr) 



Xr Xr 



XI XI 

s 2 



ab_ 
2 



XIX, 



~2 



(54) 



86 



RAILROAD CONSTRUCTION. 



§77. 



The true volume, according to the prismoidal formula, of a 
length of the road measured in this way will be 



r xi'x/ ah / xi' + xi'' x/ + x/' 1 ah\ 
L s "2"^ V 2 2 s 2 / 



x^x/' 



ah 
~2 



If computed by averaging end areas, the approximate volume 
will be 



Fj^/a:/ ab xi"xr" ah\. 



Subtracting this result from the true volume, we obtain as the 
correction 



Correction = — (x/' —a:/') (^/ —:i:/0 



{^^) 



This shows that if the side distances to either the right or 
left are equal at adjacent stations the correction is zero, and 
also that if the difference is small the correction is also small 
and very probably within the limit of accuracy obtainable by 
that method of cross-sectioning. In fact, as has already been 
shown in the latter part of § 75, it will usually be a useless 




Fig. 50. 



refinement to compute the prismoidal correction when the 
method of cross-sectioning is as rough and approximate as this 
method generally is. 

77. Equivalent level sections. These sloping " two-level " 
sections are sometimes transformed into " level sections of equal 



§ 77. EARTHWORK. 87 

area," and the volume computed by the method of level sections 
(§ 74). But the true volume of a prismoid with sloping ends 
does not agree with that of a prismoid with equivalent bases and 
level ends except under special conditions, and when this method 
is used a correction must be applied if accuracy is desired, 
although, as intimated before, the assumption that the sections 
have uniform slopes will frequently introduce greater inaccuracies 
than that of this method of computation. The following dem- 
onstration is therefore given to show the scope and limitations 
of the errors involved in this much used method. 

In Fig. 50, let d^ be the center height which gives an equiva- 
lent level section. The area will equal (aH-di)^s — ^ , which 

. , , , . ' i na ^i^r ah h 

must equal the area given m § 76, ~~cr' ^^o~' 

o 

or a-\-d^= . (56) 

To obtain c?, directly from notes, given in terms of d and a:, 

we may substitute the values of xi and Xr given in § 73, which 

gives 

, / , 7x tan ^ a-{-d 

a + d^ = {a + d) ^ •= . (57) 

V tan^ /5— tan^ a v 1 — s^ tan^ a 

The true volume of the equivalent section may be repre- 
sented by 

From this there should be subtracted the volume of the 
"grade prism" under the roadbed to obtain the volume of the 
cut that would be actually excavated, but in the following com- 
parison, as well as in other similar comparisons elsewhere made, 
the volume of the grade prism invariably cancels out, and so for 
the sake of simplicity it will be disregarded. This expression 
for volume may be transposed to 



Is Vxi'x/ , . / 




88 RAILROAD CONSTRUCTION. § 78. 

The true volume of the pnsmoid with sloping ends is (see § 76) 



['-^+<(-n^)C-^)i)-'-^]- 



The difference of the two volumes 



6s 
6s 



{xi'Xr -VXi^'x/ +Xl'Xr" ■\-Xi" Xr' —XiXr' 

-2\/x,'Xr'Xi"Xr" -x{'Xr") 
(s/xT^-V^i^'y - (58) 



This shows that '^equivalent level sections'^ do not in 
general give the true A^olume, there being an exception Avhen 
xi'xr" =^X}"xr , This condition is fulfilled when the slope is 
uniform, i.e.. when a' = a" . When this is nearly so the error 
is evidently not large. On the other hand, if the slopes are in- 
clined in opposite directions the error may be very considerable, 
particularly if the angles of slope ate also large. 

78. Three-level sections. The next method of cross -section- 
ing in the order of complexity, and therefore in the order of 




Fig. 51. 



accuracy, is the method of three-level sections. The area of 
the section is \{a^d){wT-^wi) — ---, which may be written 



II 



§ 78. EARTHWORK. ' 89 

^(a -^ d)w — -- J in which iv=Wr + itH If the volume is com' 
puted by averaging end areas, it will equal 

--l(^a^d')w' -ab-\-\a^d'')w'' -ah'\ (59) 

If we divide by 27 to reduce to cubic yards, we have, when 
Z = 100, 

Vol i,,., . )=|t(a + ^0^'-Ma64-ff(a + J")ty''— |fa6 
For the next section 

Vol (,^ . . _,=ff(a + d'')w;''-|fa&+||(a + d'")^'"'-tfa& 
For a partial station length compute as usual and multiply 

result by — '- — The prismoidal correction may bo 

obtained by applying Eq 46 to each side in turn For the left 
side we have 

~A(a^d')-{a^d'')\ivi'' -wi'), which equals 

l-{d'-d"){wr-wi'^. 



For the right side we have, similarly, 

l^(^d'-d"){Wr"-Wr')^ 

The total correction therefore equals 

Reduced to cubic yards, and with Z = 100, 

Pris. Corr.=||(cZ'-0(^''-i^'). • . ^ (60) 

When this result is compared with that given in Eq. ^^ there 
is an apparent inconsistency. If two-level ground is considered 
as but a special case of three -level ground, it would seem as if 



90 RAILROAD CONSTRUCTION. § 78. 

the same laws should apply If, in Eq. 55, Xr =Xr\ and xi" 
is different from .r/, the equation leduces to zero; but in this 
case d' would also be different from d" \ and since xi' -^Xr 
would =^w' , and xi" ^ x/' =w" in Eq. 60, w" —w' would not 
equal zero and the correction would be some finite quantity and 
not zero. The explanation lies in the difference in the form 
and volume of the prismoids, according to the method of the 
formation of the warped surfaces If the surface is supposed to 
be generated by the locus of a line moving parallel to the ends 
as plane directors and along two straight lines lying in the side 
slopes, then xi}^^^^- will equal \(xi' -\-Xi"), and a:r"^'<i- will equal 
J(a:/+x/0> but the profile of the center line will not be straight 
and 6/'»i^^- will not equal \{d'-^d"). On the other hand, if the 
surfaces be generated by two lines moving parallel to the ends 
as plane directors and along a straight center line and straight 
side lines Ivirig in the slopes, a warped surface w^ill be generated 
each side of the center line, which will have uniform slopes on 
each side of the center at the two ends and nowhere else This 
shows that when the upper surface of earthwork is warped (as 
it generally is), two-level ground should not be considered as a 
special case of three-level ground. This discussion, however, 
is only valuable to explain an apparent inconsistency and error. 
The method of two-level ground should only be used when 
such refinements as are here discussed are of no importance as 
affecting the accuracy. 

An example is given on the opposite page to illustrate the 
method of three-level sections. 

In the first column of yards 

210=|l(a + rf)t/;=||X7.3X31.1; 
507, 734, etc., are found similarly; 
595=210-61+507-61; 
448 = iVo(507-61 +734-61); 
602 =yVo (734 -61 +392 -61); 
449-392-61+179-61. 

For the prismoidal correction, 

-20 = |-K^'-(;'0(^''-'w^O=fi(2.6-8.1)(42.8-31.1) 
-rr(-5.5)( + 11.7). 

For the next line, -3 = 5^Vir[fT(-2.6)( + S.7)], and similarly 
for the rest. The " F*' in the columns of center heights, as well 



§78. 



EARTHWORK. 



91 



> 


* 


1 

1 




Tf 


•rr 


to 


CO 


CD 
1—1 






11 




+ 


+ 


+ 


-f- 


+ 






















y—^ 


































CO 


1-1 


CO 


CD 


CI 


1—1 








4- 


4- 


+ 


+ 


+ 




- 




o 


















;::-! 


















t. 


1 














1 "^ 


CO 


y-^ 


a 


rt^ 








1 

1 ^ 


00 


CO 


t-* 


. 00 








h 


1 T-H 

1 


T— I 


(M 


rH 


























e09 


cr '-<' 




o 


CO 


T— I 


CO 


r- 






.± fci 




<N 




rH 


rH 


•TTl 




CQ 


fc. O 








, 


. 


r 






PkO 




1 


1 


1 




1 


* 


O 
















o 


"g 




l> 


1> 


Tt^ 


1-1 






g 








• 






o 


o 


1 




T— 1 - 


00 


CO 


iQ 




CJ 


o 


1 




T-4 




r-^ 


1— 1 




, 








+ 


+ 


1 


1 




> 


.2 














t 


% 




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c 


P 


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? 


u 


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LO 


c^ 


Tfl 


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1 


1 


+ 


+ 




rCS 


















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th ^- 


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IS 


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r^ 


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CI 


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c--i 


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T— ' 


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1—1 


00 


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g s 


> 


(2 




CO 


-* 


lO 


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-^ 


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00 


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1—< 


^ 


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H 




+ 


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to 


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1—1 


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cq 


fe. 


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3 


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00 


TfH* 


tH 


d 


cq 


l-_ 








^ 


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^ 


Tt^' 


^ 


cq 


1—1 


O^^^ 








.^J 


fe. 


Oi 


fe.'r^ 


t*, 


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fe. 


o 


Bs 


t^ 




«*-! 


CO 




ooi • 


(N 




o 




00 




. 






^ 




(M 


•:o 




b- 




00* 




lO 


^ 






1-3 


d 


<N 


lOICO 


c 


co 


Tf 


CI 


iO rH 


CC 










T-l 1 


CI 


I— 1 
























_d 


•*« 




. 












(U 


i> 






Gs 


fe. 


fe. 


fe, 


fe. 


t:! 


Tii 






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l> 


^ 


1^ 


■^ 


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(N 


00 


d 


CO 


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V t 


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rH 








CO 
II 


§ 1 


1^ 


CO 


o 


05 


o 


n 


•^ 


tH 


»-t 


Tjl 


rH 


CI 


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o o 


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lO r^ 




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^ 55 


C 


CI C^ 



92 KAILROAD CONSTRUCTION. § 79. 

as in the columns of ''right'' and "left," are inserted to indicate 

fill for all those points. Cut would be indicated by '' C" 

25 
79. Computation of products. The quantities ~{a-\-d)w 

25 
and ^«& represent in each case the product of two variable 

terms and a constant. These products are sometimes obtained 
from tables which are calculated for all ordinary ranges of the 
variable terms as arguments. A similar table computed for 

25 

'—{d'—d''){w'^—w^) will assist similarly in computing the 
oi 

prismoidal correction. Prof. Charles L. Crandall, of Cornell 
University, is believed to be the first to prepare such a set of 
tables, which were first published in 1886 in "Tables for the 
Computation of Railway and Other Earthwork." Another 
easy method of obtaining these products is by the use of a slide- 
rule. A slide-rule has been designed by the author to accom- 
pany this volume.* It is designed particularly for this special 
work, although it may be utilized for many other purposes for 
which slide-rules are valuable. To illustrate its use, suppose 
(a + cZ) =28.2, and 1^ = 62.4; then 

25, ,, 28.2X62.4 
--(a + c^)u.= — ^-3— . 

Set 108 (which, being a constant of frequent use, is specially 
marked) on the sliding 'scale {B) opposite 282 on the other scale 
(A), and then opposite 624 on scale B Avill be found 1629 on 
scale .4, the 162 being read directly and the 9 read by estima- 
tion. Although strict rules may be followed for pointing off 
the final result, it only requires a very simple mental calculation 
to know that the result must be 1629 rather than 162.9 or 
16290. For products less than 1000 cubic yards the result 
may be read directly from the scale; for products between 1000 
and 5000 the result may be read directly to the nearest 10 

* The first edition of this book was octavo, and a pasteboard slide-rule, 
especially marked, accompanied each volume. Cutting down the size of 
^he pages to ''pocket size *" prevents the incorporation of the rule with the 
]iresent edition. Any slide-rule with a logarithmic unit 22} inches long will 
do equally well provided that the 108 mark is special!}' distinguished for 
ready use in computing the volume and that the 324 mark is similarly 
distinguished for use in computing the prismoidal correction. 



II 



§ 80. EARTHWORK. 93 

yards, and the tenths of a division estimated. Between 5000 and 
10000 yards the result may be read directly to the nearest 20 
yards, and the fraction estimated; but prisms of such volume 
will ne^'er be found as simple triangular prisms — at least, an as- 
sumption that any mass of ground was as regular as this A^'ould 
probably involve more error than would occur from faulty esti- 
mation of fractional parts. Facilities for reading as high as 
, 10000 cubic yards would not have been put on the scale ex- 
cept for the necessity of finding such products as |f(9.1X9.5), 
for example. This product would be read off from the same 
part of the rule as |f (91X95). In the first case the product 
(80.0) could be read directly to the nearest .2 of a cubic yard, 
which is unnecessarily accurate. In the other case, tlie p^-od- 
uct (8004) could only be obtained by estimating /q of a division. 
The computation for the prismoidal correction may be made 
similarly except that the divisor is 3.24 instead of 1.08. For 

example, | f (5.5 X 11.7) =5:^^^^iJ^. Set the 324 on scale B 

(also specially marked like 108) opposite 55 on scale A, and 
proceed as before. 

8o. Five-level sections. Sometim.es the elevations over each 
edge of the roadbed are obser\'ed when cross-sectioniog. These 
are distinctively termed ^'five-level sections." If the center, 
the slope-stakes, and one intermediate point on each side {not 
necessarily over the edge of the roadbed) are observed, it is 
termed an 'irregular section." The field-work of cross-section- 
ing five-level sections is no less than for irregular sections with 
one intermediate point; the computations, although capable of 
peculiar treatment on account of the location of the intermediate 
point, are no easier, and in some respects more laborious; the 
cross-sections obtained will not in general represent the actual 
cross-sections as truly as when there is perfect freedom in locat- 
ing the intermediate point; as it is generally inadvisable or un- 
necessary to employ five-level sections throughout the length of 
a road, the change from one method to another adds a possible 
element of inaccuracy and loses the advantage of uniformity of 
method, particularly in the notes and form of computations. 
On these accounts the method will not be further developed, 
except to note that this case, as well as any other, may be 
solved by dividing the whole prismoid into triangular prismoids, 
computing the volume by averaging end areas, and computing 



94 



RAILROAD CONSTRUCTION. 



§81, 



the prismoidal correction by adding the computed corrections 
for each elementary triangular prismoid. 

8i. Irregular sections. In cross-sectioning irregular sections, 
the distance from the center and the elevation above "grade" 
of every "break" in the cross-section must be observed. The 
area of the irregidar section may be obtained by computing the 
area of the trapezoids (/ire, in Fig. 44) and subtracting the two 
external triangles. For Fig. 44 the area would be 

hi + ki. ki + d ,d + jr ,jr + kr. . 

-j—i^i-yi) + ^-yi+ ~2^^r+~^(yr-Zr) 

,kr + hr, . hi/ h\ hr/ h\ 




Fig. 44. 



Expanding this and collecting terms, of which many will 
cancel, we obtain 



Area = — xiki+yi(d—hi) +Xrkr-\-yr(ir'-hr) 



+ Zr(d-k,) + Ahl + hr) 



. (61) 



An examination of this formula will show a perfect regu- 
larity in i<s formation which will enable one to write out a 
similar formula for any section, no matter how irresjular or how 



one 



§ 82. EARTHWORK. 95 

many points there are, ^vithout any of the preliminary work. 
The formula may be expressed in words as follows: 

Area equals one-holf the sum of products: obtained as follows : 

the distance to each slope-stake times the height above grade of 
the point next inside the slope-stake; 

the distance to each intermediate point in turn times the height of 
the point just inside minus the height of the point just outside; 

ftnaUy, one-half the width of the roadbed times the sum of the 
slope-stake heights. 

If one of the sides is perfectly regular from center to slope- 
stake, it is easy to show that the rule holds literally good. The 
^' point next inside the slope-stake" in this case is the center; 
the intermediate terms for that side vanish. The last term 
must always be used. The rule holds good for three-level sec- 
tions, in which case there are three terms, Avhich may be reduced 
to two. Since these two terms arc both variable quantities for 
each cross- section, the special method, given in § 78, in which 

term ( -77 ) is a constant for all sections, is preferable. In 

the general method, each intermediate "break" adds another 
term. 

82. Volume of an irregular prismoid. If there is a break at 
one cross- section which is not represented at the next, the ridge 
(or hollow) implied b}^ that break is supposed to "vanish" at 
the next section. In fact, the volume will not be correctly 
represented unless a cross- sectipn is taken at the point where 
the ridge or hollow "vanishes" or "runs out " To obtain the 
true prismoidal correction it is necessary to observe on the 
ground the place where a break in an adjacent section, which 
is not represented in the section being taken, runs out. For 
example, in Fig. 52, the break on the left of section A" , at a 
distance of yi" from the center, is observed to run out in section 
A/ at a distance of yi' from the center. The volume of the 
prismoid, computed by the prismoidal formula as in § 70, will 
involve the midsection, to obtain the dimension of which would 
require a laborious computation. A simpler process is to com- 
pute the volume by averaging end areas as in § 81 and apply a 
prismoidal correction. To do this write out an expression for 
each end area similar to that given in Eq. 61. The sum of 

these areag times ^ gives the approximate volume. As before, 



96 



RAILROAD CONSTRUCTION. 



§83. 



for partial station lengths, multiply the result by — ^^^ . 

There will be no constant sub tractive term, ffa/j, as in § 78. 
The true prismoidal correction may be computed, as in § 83, or 
the following approximate method may be used: Consider the 
irregular section to be three-level ground for the purpose of 




Fig. 52. 



computing the correction only. This has the advantage of less 
labor in computation than the use of the true prismoidal correc- 
tion, and although the error involved may be considerable in 
individual sections, the error is as likely to be positiA^e as nega- 
tive, and in the long run the error will not be large and generally 
will be much less than would result by the neglect of any pris- 
moidal correction. 

83. True prismoidal correction for irregular prismoids. As 
intimated in § 82, each cross-section should be assumed to have 
the same number of sides as the adjacent cross-section when 
computing the prismoidal correction. This being done, it per- 
mits the division of the whole prismoid into elementary triangu- 
lar prismoids, the dimensions of the bases of which being given 
in each case by a vertical distance above grade line and by the 
horizontal distance between two adjacent breaks. The summa- 
tion of the prismoidal corrections for e.ach of the elementary 
triangular prismoids will give the true prismoidal correction. 
Assuming for an example the cross-section of Fig. 44, with a 
cross-section of the same number of sides, and with dimensions 



§ 84. EARTHWORK. 97 

similarly indicated, for the other end, the prismoidal correction 
becomes (see Eq. 46) 



w - hi")[{xi" - yn - w - yi')] + (h' - kmw - yn- (x/ - yn] 

■Viki' -kniyi" -yi')-V{d' -d''){yr -yi') + {d' -d"){zr" -Zr') 
+ ikr'-kr'')[(y/'-z/') - (y/-z/)] 

b \ / . h 



-(v-.n[(«"-i)-(.'-i)] 

-(V-V')[(x/'-|)-(./-f)]~ 



Expanding this and collecting terms, of which many will 
cancel, we obtain 

Pris.Corr = j^[(^/' - ^I'Kh' - kn + W - 2//)[(c?' - hi') - {d" - /ii")] 

+ {Xr" -Xr'){kr' -kr") + {yr" -yr')[U/ -hr')-Ur" -hr")'\ 

+ {Zr"-Zr')[{d'-kr')~{d"-kr")']\ (62) 



By comparing this equation with Eq. 61 a remarkable coin- 
cidence in the law of formation may be seen, which enables 
this formula to be written by mere inspection and to be applied 
numerically with a minimum of labor from the computations for 
end areas, as will be shown (§ 84) by a numerical example. 
For each term in Eq. 61, as, for example, yrijr-^hr), there ii 
a correction term in Eq. 62 of the form 

Each one of these terms [?//', ?.//, U/—h/), and (//'— /i/O] has 
been previously used in finding the end areas and has its place 
in the computation sheet. The summation of the products ck^ 
these differences times a constant gives the total true prismoidal 
correction in cubic yards for the whole prismoid considered. 

The constant is the same as that computed in § 78, i.e., |J. 

84. Numerical example; irregular sections; volume with true 
prismoidal correction. (See page 98.) 

Roadbed 18 feet wide in cut; slope IJ to 1. 



98 



RAILROAD CONSTRUCTION. 



§85. 



Sta. 

19 

18 

17 

+ 42 

16 



( cut 



Center -( or 
/fill. 



0.6c 
2.3c 
7.6c 
10.2c 
6.8c 



3.6c 
14.4 

4.2c 
15.3 

8.2c 
21.3 



27.3 

8.9c 
22.4 



Left. 



/ 2.3c \ / ] .8c \ 
V8.2 ; V 60 / 



6 8c 

8.4 

10.2c 
17.4' 



\ 22.0 / 





3.2c 
5.2 

8.0c 
6.1 

12 6c 

8.2 

7.6c 
12.0 



Right. 



0.1c 



4.1 



0.4c 



4.2 


9.6 


m 


1.2c 
10.8 


m) 


4.2c 
15.3 


6.2c 


8.4c 


7.5 


21.6 


3.2c 


2.6c 



12.9 



The figures in the bracket ( i^^rr ) mean that it was noted in 

the field that the break, indicated at Sta. 17 as being 17.4 to 
the left, ran out at Sta. 16 + 42 at 22.0 to the left. By inter- 
polation between 8.2 and 27.3 the height of this point is computed 
as 12.3. The quantities in the other brackets are obtained 
similarly. These quantities are only used when the computation 
of the true prismoidal correction is desired. They are not 
needed in computing the volume by averaging end areas, nor 
are they used at all if the prismoidal correction is to be obtained 
by assuming (for this purpose) the ground to be three-level ground. 

In the tabular form on page 99 the figures within the braces 
( — . — ) are not used in computing the volume, but are only 
used to obtain the differences of widths or heights with which 
to compute the true prismoidal correction. It may be noted, 
as a check, that the volume, computed from these figures in the 
braces, is the same as that computed from the other figures 
The figures within each brace (or bracket) constitute a group 
which must be used in connection with a group which has the 
same number of points, on the same side of the center, in the 
next cross-section, previous or succeeding. In the column of 
"Yards" under ''True pris. corr.," we have, for example, 
(-5)=t1)%(-7 + 0-8 + 3). 

85. Volume of irregular prismoid, with approximate prismoidal 
correction. If the prismoidal correction is obtained approxi- 



86. 



EARTHWORK. 



99 



VOLUME OF IRREGULAR PRISMOID, WITH TRUE PRISMOIDAL 

CORRECTION. 











True pris. corr. 


Sta. 


Width. 


Height. 


Yards. 










w"-w' 


h'-h" 


Yards. 




^['22.4 
^ 12.0 


7.6 


158 


1 










— 2.1 


— 23 










16 


4.1 ^ 


3.2 


40 


i 










4.2 


16 


1 










9.0 


11.5 


96 


1 










T r27.3 

^ 8.2 


12.6 


319 




+ 4.9 


—5.0 


-7 




— 2.0 


— 15 




-3.8 


-0.1 







\27.3 


12.3 












+ 42 


L^22.0 
( 8.2 


0.4 
— 2.1 














21.6"p 
7.5 ^ 


6.2 


124 




+ 8.7 


-3.0 


-8 




1.8 


13 




+ 3.4 


+ 2.4 


+ 3 




9.0 


20.6 


172 


378 






(-5) 




r21.3 


10.2 


201 


1 


-6.0 


+ 2.1 


-4 




L 17.4 


— 0.2 


— 3 




-4.6 


+ 0.6 


-1 




L 6.1 


— 2.6 


— 14 




—2.1 


+ 0.5 





17 


15.3It. 

8.or^ 


5.8 




1 


—6.3 


+ 0.4 


-1 




3.4 






+ 0.5 


-1.6 







15.3]R 


7.6 


107 












9.0 


12.4 


103 


584 






(-3) 




ri5.3 


6.8 


95 




-6.0 


+ 3.4 


-6 




L 8.4 


— 1.0 


— 7 


1 


-9.0 


+ 0.8 


-2 




L 5.2 


— 4.5 


— 22 




-0.9 


+ 1.9 


-1 


18 


10.8]R 
10.8/ x> 
3.6f^ 


2.3 
1.9 


23 




-4.5 


+ 5.3 


-7 




1.1 














9.0 


5.4 


45 


528 






(-16) 




L[14.4 


0.6 


8 


i 










(14.4 . 


2.3 




i 


-0.9 


+ 4.5 


-1 




l] 8.2 


— 1.8 




1 
1 


-0.2 


+ 0.8 





19 


( 6.0 


— 1.7 




1 


+ 0.8 


-2.8 


-1 




4.2 ^ 


0.1 


1 


1 


-1.2 


+ 1.8 


-1 




0.2 


1 




+ 0.6 


+ 0.9 







9.0 


4.0 


33 


177 






(-3) 



Approx. vol. = 1667 
True pris. corr = — 27 



-27 



True volume = 1640 cubic yards 

mately, by the method outHned in § 82, the process will be as 
shown in the tabular form on page 100. Not only is the 
numerical work considerably less than the exact method, but the 
discrepancy in cubic yards is almost insignificant. 

86. Illustration of value of approximate rules. The tabula- 
tion on page 100 shows that when the volume of an irregular 
prismoid is computed by averaging end areas and is corrected 

L.oFC. 



100 



RAILROAD CONSTRUCTION. 



§86. 



VOLUME OF IRREGULAR PRISMOID, WITH APPROXIMATE 
PRISMOIDAL CORRECTION. 



Sta. 


W'th 


H'ght 


Ya 

158 

-23 

40 

16 

96 


rds. 


Cen. 
Height. 


Total 
width 


d'-d" 


w"-w' 


Approx. 
pris. corr. 


16 


22.4 

12.0 

12.9 

4.1 

9.0 


7.6 

-2.1 

3.2 

4.2 

11.5 


+ 6.8 


35.3 








+ 42 


27.3 

8.2 

21.6 

7.5 
9.0 


12.6 

-2.0 

6.2 

1.8 

20.6 


319 

-15 

124 

13 

172 


378 


+ 10.2 


48.9 


-3.4 


+ 13.6 


-14 

(-6) 


17 


21.3 
17.4 

6.1 
15.3 

9.0 


10.2 

-0.2 

-2.6 

7.6 

12.4 


201 

- 3 

-14 

107 

103 


584 


+ 7.6 


36.6 


+ 2.6 


-12.3 


-10 

(-6) 


18 


15.3 
8.4 
5.2 

10.8 
9.0 


6.8 

-1.0 

-4.5 

2.3 

5.4 


95 

- 7 

-22 

23 

45 


528 


+ 2.3 


26.1 


+ 5.3 


-10.5 


-17 

(-17) 


19 


14.4 
9.6 
4.2 
9 


0.6 
0.1 
0.2 
4.0 


8 

1 

1 

33 


177 


+ 0.6 


24.0 


+ 1.7 


-2.1 


-1 

(-1) 



Approx. volume = 1667 
Approx. pris. corr. = — 30 



30 



Corrected volume = 1637 cubic yards 

by considering the ground as three-level ground {for the pur- 
poses of the correction only), the error for the different sections 



Sections. 



16 16 + 42 

16 + 42. .17 

17 18 

18 19 



. 


^ w 


Ui 


o 


O G M 


o ■. 


H 


^ "bC c3 


o.^ 


JB 


X 2 h 


§^h 


> 


Appro 

by ave 

end a 


, Differe 
' true 

Ox CO 


373 


378 


581 


584 


- 3 


512 


528 


-16 


174 


177 


- 3 

— 27 


1640 


1667 



Approx. pris. 

corr. on basis 

of three-level 

ground. 


Error. : 


— 6 

— 6 

— 17 

— 1 


— 1 
-3 

-1 
+ 2 


-30 


-3 



>1 



?^ ra aj 



396 
577 
463 
147 



1583 



Error. 



-23 

+ 4 
+ 49 

+ 27 



+ 57 



is sometimes positive and sometimes negative, and in this case 
amounts to only 3 yards in 1640 — less than 



i of 1%. 



If the 



§87. 



EARTHWORK. 



101 



prismoidal correction had been neglected, the error would have 
been 27 yards — nearly 2%. The approximate results are here 
too large for each section — as is usualh^ the case. If points 
between the center and slope-stakes are omitted and the volume 
computed as if the ground were three-level ground, the error is 
quite large in individual sections, but the errors are both posi- 
tive and negative and therefore compensating. 

87. Cross-sectioning irregular sections. The prismoids con- 
sidered have straight lines joining corresponding points in the 
two cross-sections. The center line must be straight between 
two cross-sections. If a ridge or valley is found lying diago- 
nally across the roadbed, a cross-section must be interpolated at 
the lowest (or highest) point of the profile. Therefore a ^' break" 
at any section cannot be said to run out at the other section on 
the opposite side of the center. It must run out on the same 
side of the center or possibly at the center. Very frequently 
complicated cross-sectioning may be avoided by computing the 
volume, by some special method, of a mound or hollow when 
the ground is comparatively regular except for the irregularity 
referred to. 

88. Side-hill work. When the natural slope cuts the roadbed 
there is a necessitv for both cut and fill at the same cross-section. 



^^5^5^ 




Fig. 53. 



When this occurs the cross-sections of both cut and fill are often 
so nearly triangular that they may be considered as such without 
great error, and the volumes may be computed separately as 
triangular prismoids without adopting the more elaborate form 
of computation so necessary for complicated irregular sections. 
When the ground is too irregular for this the best plan is to 



102 



KAILROAD CONSTRUCTION. 



§88. 



follow the uniform system. In computing the cut, as in Fig, 53. 
the left side would be as usual; there would be a small center 
cut and an ordinate of zero at a short distance to the right of the 
center. Then, ignoring the fill, and applying Eq. 61 strictly, 
we have two terms for the left side, one for the right, and the 
term involving i&, which will be ^hhi in this case, since hr = 0, 
and the equation becomes 

Area = i[xiki + yi{d —hi) + Xrd + ihhi]. 

The area for fill may also be computed by a strict application 
of Eq. 61, but for Fig. 54 all distances for the left side are zero 
and the elevation for the first point out is zero, d also must be 




Fig. 54. 



considered as zero. Following the rule, § 81, literally, the equa- 
tion becomes 

Area(Fiii) =i[xrkr-\-yr(o—hr)+Zr(o—kr)+ih(o + hr)l 



which reduces to 



i[Xrkr—yrhr — Zrkr + ihhr], 



(Note that Xr, hr, etc., have different significations and values 
in this and in the preceding paragraphs.) The ^'terminal 
pyramids'' illustrated in Fig. 40 are instances of side-hill work 
for very short distances. Since side-hill work always implies 
both cut and fill at the same cross-section, whenever either the 
cut or fill disappears and the earthwork becomes wholly cut or 
wholly fiU, that point marks the end of the '^ side-hill work," 
and a cross-section should be taken at this point. 



§ 89. EARTHWORK. 103 

89. Borrow-pits. The cross-sections of borrow-pits will vary 
not only on account of the undulations of the surface of the 




uihihiiimnniiiiiiniihWwnniii. ' innjiniiiif' 

Tig. 55, 

ground, but also on the sides, according to whether they are 
made by widening a convenient cut (as illustrated in Fig. 55) 
or simply by digging a pit. The sides should always be prop- 
erly sloped and the cutting made cleanly, so as to avoid un- 
sightly roughness. If the slope ratio on the right-hand side 
(Fig. 55) is s, the area of the triangle is \sm}. The area of the 
section is \[ug-\-{g + h)v^-Qi + f)x-^{j + h)y-^{k-]-m)z — sm'^]. If 
all the horizontal measurements were referred to one side as 
an origin, a formula similar to Eq. 61 could readily be devel- 
oped, but little or no advantage would be gained on account of 
any simplicity of computation. Since the exact A^olume of the 
earth borrowed is frequently necessary', the prismoidal correc- 
tion should be computed; and since such a section as Fig. 55 
does not even approximate to a three-level section, the method 
suggested in § 82 cannot be employed. It will then be neces- 
sary to employ the exact method, § 83, by dividing the volume 
into triangular prismoids and taking the summation of their 
corrections, found according to the general method of § 71. 

90. Correction for curvature. The volume of a solid, gen- 
erated by revolving a plane area about an axis lying in the 
plane but outside of the area, equals the product of the given 
area times the length of the path of the center of gravity of the 
area. If the centers of gravity of all cross-sections lie in the 
center of the road, where the length of the road is measured, 
there is absolutely no necessary correction for curvature. If all 
the cross-sections in any given length were exactly the same and 
therefore had the same eccentricity, the correction for curvature 
would be very readily computed according to the above prin- 
ciple. But when both the areas and the eccentricities vary 
from point to point, as is generally the case, a theoretically exact 



iSt^ 






104 RAILROAD CONSTRUCTION. § 90. 

solution is quite complex, both in its derivation and application 
Suppose, for simplicity, a curved section of the road, of uniform 
cross-sections and with the center of gravity of every cross- 
section at the same distance e from the center line of the road 
The length of the path of the center of gravity will be to the 
length of the center line as R±e : R. Therefore we have 

R:te 
True vol.: nominal vol. :: R±e ' R. .*. True voL=lA — ^— for 

a volume of uniform area and eccentricity. For any other area 

R±e^ 
and eccentricity we have, similarly, True vol.' = IA' — ^— . This 

shows that the effect of curvature is the same as increasing (or 

diminishing) the area by a quantity depending on the area and 

eccentricity, the increased (or diminished) area being found by 

R + e 
multiplying the actual area by the ratio —75—. This being 

independent of the value of I, it is true for infinitesimal lengths. 
If the eccentricity is assumed to vary uniformly between two 
sections, the equivalent area of a cross-section located midway 



(-4^) 



between the two end cross-sections would be A,„ „ 

K 

Therefore the volume of a solid which, when straight, w^ould be 
— (A' + 4A»i4-A''), would then become 



True vol. =-—, 
6R 



A'*(E±60+4.4.,,(/^±^^') +A-(i^±o1. 



Subtracting the nominal volume (the true volume w4ien the 
prismoid is straight), the 



Correction -±^\ ( A ' + 2 A ,0 e' + (2A ^ + A '0 e' 



(63) 



Another demonstration of the same result is given by Prof. 
C. L. Crandall in his ^'Tables for the Computation of Railway 
and other Earthwork," in which is obtained by calculus methods 
the summation of elementary volumes ha\'ing variable areas 
with variable eccentricities. The exact application of Eq. €3 
requires that Am be known, which requires laborious computa- 



§91 



EARTHWORK, 



105 



tions, but no error worth considering is involved if the equation 
is written approximately 



Curv. corr. = —AA'e'^A"e"), 



(64) 



which is the equation generally used. The approximation con- 
sists in assuming that the difference between A^ and A^n equals 
the difference between /!,» and A" but with opposite sign. The 
error due to the approximation is ah\'ays utterly insignificant. 

91. Eccentricity of the center of gravity. The determination 
of the true positions of the centers of gravity of a long series of 
irregular cross-sections would be a very laborious operation, 
but fortunately it is generally sufficiently accurate to consider 
the cross-sections as three-level ground, or, for side-bill work, to 




\i / 



Fig. 56. 

be triangular, for the purpose of this correction. The eccentricity 
of the cross-section of Fig. 56 (including the grade triangle) may 
be written 



e = 



(a-\-d)xiXi 
2 3 


(a-\-d)XrXr 

2 3 


1 X?-Xr' 


{a-^d)xj 


, (a + d)xr 


3 Xi-i-Xr 



=i(x, 



■Xr). 



(65) 



The side toward xi being considered positive in the above 
demonstration, if Xr>xi, e Avould be negative, i.e., the center 
of gravity would be on the right side. Therefore, for three-level 



106 RAILROAD CONSTRUCTION. § 91. 

ground, the correction for curvature (see Eq.64) may be written 

Correction = J^[A\x{ -x/) +A"{xi"-x;')l 



Since the approximate volume of the prismoid is 
l(A+A')=^A^ + ^A'^ = V^ + W^, 

in which F' and V represent the number of cubic yards corre- 
sponding to the area at each station, we may write 

Corr. in cub. yds. = Al^W -^/) + V'^x^ -x/')l . (66) 

It should be noted that the value of e, derived in Eq. 65, is 
the eccentricity of the whole area including the triangle under 
the roadbed. The eccentricity of the true area is greater than 
this and equals 

true area + hah 

e X — ; ^~ = e^. 

true area 

The required quantity (A'e' of Eq. 64) equals true areaXei 
which equals (true area-\-^ab)Xe. Since the value of e is very 
simple, while the value of e^ would, in general, be a complex 
quantity, it is easier to use the simple value of Eq. 65 and add 
^ab to the area. Therefore, in the case of three-level ground 
the subtractive term ffa& (§ 78) should not be subtracted in 
computing this correction. For irregular ground, when com- 
puted by the method gi\^en in §§81 and 82, which does not 
involve the grade triangle, a term ffa5 must be added at every 
station when computing the quantities V^ and F" for Eq. 66. 

It should be noted that the factor 1^37^, which is constant 
for the length of the curve, may be computed with all necessary 
accuracy and without resorting to tables by remembering that 

5730 



degree of curve* 



Since it is useless to attempt the computation of railroad 
earthwork closer than the nearest cubic yard, it will frequently 



» 



91. 



EARTHWORK. 



107 



be possible to write out all curvature corrections by a simple 
mental process upon a mere inspection of the computation sheet. 
Eq. 66 shows that the correction for each station is of the form 

V(Xi-Xr) 



SR 



3R is generally a large quantity — for a 6° curve 



it is 2865. {xi—Xr) is generally small. It may frequently be 
seen by inspection that the product V(xi—Xr) is roughly twice 
or three times SR, or perhaps less than half of 37^, so that the 
corrective term for that station may be written 2, 3; or cubic 
yards, the fraction being disregarded. For much larger absolute 
amounts the correction must be computed with a correspondingly 
closer percentage of accuracy. 

The algebraic sign of the curvature correction is best deter- 
mined by noting that the center of gravity of the cross-section is 
on the right or left side of the center according as Xr is greater 
or less than xi, and that the correction is positive if the center of 
gravity is on the outside of the curve, and negative if on the 
inside. 

It is frequently found that xi is uniformly greater (or uni- 
formly less) than Xr throughout the length of the curve. Then 
the curvature correction for each station is uniformly positive or 
negative. But in irregular ground the center of gravity is apt 




Fia. 57. 



to be irregularly on the outside or on the inside of the curve, 
and the curvature correction will be correspondingly positive or 
negative. If the curve is to the right, the correction will be 
positive or negative according a,s (xi—Xr) is positive or negative: 
if the curve is to the left, the correction will be positive or nega- 



108 RAILROAD CONSTRUCTION. § 92. 

tive according as (xr—xi) is positive or negative. Therefore 
when computing curves to the right use the form (xi—Xr) in 
Eqs. 66 and 68; when computing curves to the left use the form 
(xr—xi) in these equations; the algebraic sign of the correction 
will then be strictly in accordance with the results thus obtained. 
92. Center of gravity of side-hill sections. In computing the 
correction for side-hill w^ork the cross-section would be treated 
as triangular unless the error involved would evidently be too 
great to be' disregarded. The center of gravity of the triangle 
lies on the line joining the vertex with the middle of the base 
and at J of the length of this line from the base. It is therefore 
equal to the distance from the center to the foot of this line plus 
J of its horizontal projection. Therefore 



h 

~4 


Xr XI 

2"^3 


h Xr 

~ 12 6 


h XI Xr 

""6"^3"3 

• 




1 


}+(-- 


-Xr) 




3 





(67) 



By the same process as that used in § 91 the correction equation 
may be written 



Corr. in cub. y^^' = ^\y'{\'^ ('^^' - ^r^) + V-(^ + (xi- -x/') )]. 



(68) 



It should be noted that since the grade triangle is not used in 
this computation the volume of the grade prism is not involved 
in computing the quantities F' and F''. 

The eccentricities of cross-sections in side-hill work are never 
zero, and are frequently quite large. The total volume is gen- 
erally quite small. It follows that the correction for curvature 
is generally a vastly larger proportion of the total volume than 
in ordinary three-level or irregular sections. 

If the triangle is wholly to one side of the center, Eq. 67 can 
still be used. For example, to compute the eccentricity of the 
triano-le of fill, Fig. 57, denote the two distances to the slope- 



§ 93. EARTHWORK. 109 

stakes by yr and —yi (note the minus sign). Applying Eq. 67 
literally (noting that — must here be considered as negative in 
order to make the notation consistent) we obtain 



which reduces to 



1 



■] 



\-\-yi^Vr 1 (69) 



As the algebraic signs tend to create confusion in these 
formulce, it is more simple to remember that for a triangle 
lying on hoth sides of the center e is always numerically equal 

to^-] --^{xi^x^^ L and for a triangle entirely on one side, e is 

numerically equal to — ^ + ^^^ numerical swfn of the two dis- 
tances out]. The algebraic sign of e is readily determinable as 
in § 91. 

93. Example of curvature correction. Assume that the fill in 

§ 78 occurred on a 6° curve to the right, -— = . The 

quantities 210, 507, etc., represent the quantities F', F", etc.,. 
since they include in each case the 61 cubic yards due to the 
grade prism. Then 



F(a:z^.Tr) _ 210(22.9-8.2) ^ 3101.7 _^ 
ZR ~ 2865 2865 ~ "^ 



The sign is plus, since the center of gravity of the cross-sec- 
tion is on the left side of the center and the road curves to the 
right, thus making the true volume larger. For Sta. 18 the 
correction, computed similarly, is +3, and the correction for 
the whole section is 1+3=4. For Sta. 18 + 40 the correction 
is computed as 6 yards. Therefore, for the 40 feet, the correc- 
tion is i\j"o(3 + 6) =3.6, which is called 4. Computing the others 
similarly we obtain a total correction of + 16 cubic yards. 



110 RAILROAD CONSTRUCTION. § 94. 

94. Accuracy of earthwork computations. The preceding 
methods give the 'precise volume (except where approximations 
are distinctly admitted) of the prismoids which are supposed to 
represent the volume of the earthwork. To appreciate the 
accuracy necessary in cross-sectioning to obtain a given accuracy 
in volume, consider that a fifteen-foot length of the cross-section, 
which is assumed to be straight, really sags 0.1 foot, so that the 
cross-section is in error by a triangle 15 feet wide and 0.1 foot 
high. This sag 0.1 foot high would hardly be detected by the 
eye, but in a length of 100 feet in each direction it would make 
an error of volume of 1.4 cubic yards in each of the two pris- 
moids, assuming that the sections at the other ends were perfect. 
If the cross-sections at both ends of a prismoid were in error by 
this same amount, the volume of that prismoid would be in error 
by 2.8 cubic yards if the errors of area were both plus or both 
minus. If one were plus and one minus, the errors would 
neutralize each other, and it is the compensating character of 
these errors which permits any confidence in the results as 
obtained by the usual methods of cross-sectioning. It demon- 
strates the utter futility of attempting any closer accuracy than 
the nearest cubic yard. It will thus be seen that if an error 
really exists at any cross-section it involves the prismoids on 
hoth sides of the section, even though all the other cross-sections 
are perfect. As a further illustration, suppose that cross-sec- 
tions were taken by the method of slope angle and center depth 
(§ 73), and that a cross-section, assumed as uniform, sags 0.4 
foot in a width of 20 feet. Assume an equal error (of same 
sign) at the other end of a 100-foot section. The error of 
volume for that one prismoid is 38 cubic yards. 

The computations further assume that the warped surface, 
passing through the end sections, coincides with tke surface of 
the ground. Suppose that the cross-sectioning had been done 
with mathematical perfection; and, to assume a simple case, 
suppose a sag of 0.5 foot between the sections, which causes an 
error equal to the volume of a pyramid having a base of 20 feet 
(in each cross-section) times 100 feet (between the cross-sec- 
tions) and a height of 0.5 foot. The volume of this pyramid is 
i(20Xl00)X0.5 = 333 cub. ft. = 12 cub. yds. And yet this sag 
or hump of 6 inches would generally be utterly unnoticed, or 
at least disregarded. 

When the ground is very rough and broken it is sometimes 



§ 95. EARTHWORK. * 111 

practically impossible, even with frequent cross-sections, to 
locate warped surfaces which will closely coincide with all the 
sudden irregularities of the ground. In such cases the compu- 
tations are necessarily more or less approximate and dependence 
must be placed on the compensating character of the errors. 

95. Approximate computations from profiles. As a means 
of comparing the relative amounts of earthwork on two or 
more proposed routes Avhich ha^'e been surA^eyed by preliminary 
surveys, it Avill usually be sufficiently accurate to compare the 
areas of cutting (assuming that the cut and fill are approximately 
balanced) as shown by the several profiles The errors involved 
may be large in individual cases and for certain small sections, 
but fortunately the errors (in comparing two lines) will be 
largely compensated. The errors are much larger on side-hill 
work than when the cross-sections are comparatively level. 
The errors become large when the depth of cut or fill is very 
great. If the lines compared have the same general character 
as to the slope of the cross-sections, the proportion of side-hill 
work, and the average depth of cut or fill, the error involved in 
considering their relative A'olumes of cutting to be as the relative 
areas of cutting on the profiles (obtained perhaps by a plani- 
meter) will probably be small If the volume in each case is 
computed by assuming the sections as level, with a depth equal 
to the center cut, the error involved will depend only on the 
amount of side-hill work and the degree of the slope. If these 
features are about the same on the two lines compared, the error 
involved is still less. 

FORMATION OF EMBANKMENTS. 

96. Shrinkage of earthwork. The evidence on this subject 
as to the amount of shrinkage is very conflicting, a fact which 
is probably due to the following causes: 

1. The various kinds of earthy material act ^^er}' differently 
as respects shrinkage. There has been but little uniformity in 
the classification of earths in the tests and experiments that have 
been made. 

2. Very much depends on the method of forming an embank- 
ment (as will be shown later). Different reports have been 
based on different methods — often without mention of the 
method. 



112 ' RAILROAD CONSTRUCTION. § 96. 

3. An embankment requires considerable time to shrink to 
its final volume, and therefore much depends on the time elapsed 
between construction and the measurement of what is supposed 
to be the settled volume. 

P. J. Flynn quotes some experiments {Eng. News, May 1, 
1886) made in India, in which pits were dug having volumes of 
400 to 600 cubic feet. The material, when piled into an em- 
bankment, measured largely in excess of the original measure- 
ment — as is the universal experience. The pits were refilled 
with the same material. As the rains, very heavy in India, 
settled the material in the pits, more was added to keep the pits 
full. Even after the rainy season was over, there was in every 
case material in excess. This would seem to indicate a per- 
manent expansiorij although it is possible that the observations 
were not continued for a sufficient time to determine the final 
settled volume. 

On the contrary, notes made by Mr. Elwood Morris many 
years ago on the behavior of embankments of several thousand 
cubic yards, formed in layers b}^ carts and scrapers, one winter 
intervening between commencement and completion, showed in 
each case a permanent contraction averaging about 10%. 

All authorities agree that rockwork expands permanently 
when formed into an embankment, but tho percentages of 
expansion given by different authorities differ even more than 
with earth — varying from 8 to 90%. Of course this very large 
range in the coefficient is due to differences in the character of 
the rock. The softer the rock and the closer its similarity to 
earth, the less will be its expansion. On account of the conflict- 
ing statements made, and particularly on account of the influence 
of methods of work, but little confidence can be felt in any 
given coefficient, especially when given to a fraction of a per 
cent, but the consensus of American practice seems to average 
about as follows: 

Permanent contraction of earth about 10% 

' ' expansion of rock 40 to 60% 

These values for rock should be materially reduced, according 
to judgment, when the rock is soft and hable to disintegrate. 
The hardest rocks, loosely piled, may occasionally give even 



§97. 



EARTHWORK, 



113 



higher results. The following is given by several authors a^ 
the permanent contraction of several grades of earth: 

Gravel or sand about 8% 

Clay '' 10% 

Loam '' 12% 

Loose vegetable surface soil. . * ^ 15% 

It may be noticed from the above table that the harder and 
cleaner the material the less is the contraction. Perfectly clean 
gravel or sand would not probably change volume appreciably. 
The above coefficients of shrinkage and expansion may be used 
to form the following convenient table. 



Material. 


To make 1000 cubic 

yards of embankment 

will require 


1000 cubic yards 
measured m exca- 
vation will make 


Gravel or sand 


1087 cubic yards 

nil '• 

1186 " 
1176 " 

714 '* 

625 " 
measured in excavation 


920 cubic yards 


Clay 


900 " 


I joam 


880 


Loose vegetable soil 

Rock, large pieces 

" small " 


850 " 
1400 " 
1600 " 
of embanknient. 



97. Allowance for shrinkage. On account of the initial ex- 
pansion and subsequent contraction of earth, it becomes neces- 
sary to form embankments higher than their required ultimate 
form in order to allow for the subsequent shrinkage. As the 
shrinkage appears to be all vertical (practically), the embank- 
ment must be formed as shown in Fig. 58. The effect of shrink- 





Fig. 58. 



age should not be confounded with that of slipping of the sides, 
which is especial!}' apt to occur if the embankment is subjected 



114 KAILROAD CONSTRUCTION. § 98. 

to heavy rains very soon after being formed, and also when the 
embankments are originally steep. It is often difficult to form 
an embankment at a slope of 1 : 1 which will not slip more or 
less before it hardens. 

Very high embankments shrink a greater percentage than 
lower ones. Various rules giving the relation between shrink- 
age and height have been suggested, but they vary as badly as 
the suggested coefficients of contraction, probably for the same 
causes. As the fact is unquestionable, however, the extra 
height of the embankment must be varied somewhat as in Fig. 
59, which represents a longitudinal section of an embankment. 



Fig. 59. 

As considerable time generally elapses between the completion 
of the embankment and the actual running of trains, the grade 
ad will generally be nearly flattened down to its ultimate form 
before traffic commences, but such grades are occasionally objec- 
tionable if added to what is already a ruling grade. With some 
kinds of soil the time required for complete settlement may be 
as much as two or three years, but, even in such cases, it is 
probable that one-half of the settlement will take place during 
the first six months. The engineer should therefore require 
the contractor to make all fills about 8 to 15% (according to the 
material) higher than the profiles call for, in order that subse- 
quent shrinkage may not reduce it to less than the required 
volume. 

98. Methods of forming embankments. When the method is 
not otherwise objectionable, a high embankment can be formed 
very cheaply (assuming that carts or wheelbarrows are used) by 
dumping over the end and building to the full height (or even 
higher, to allow for shrinkage) as the embankment proceedr. 
This allows more time for shrinkage, saves nearly all the cost of 
spreading (see Item 4, § 111), and reduces the cost of roadways 



§98. 



EARTHWORK. 



115 



(Item 5). Of course this method is especially applicable when 
the material comes from a place as high as or higher than grade, 
so that no up-hill hauling is required. 

Another method is to spread it in layers two or three feet 
thick (see Fig. 60), which are made concave upwards to avoid 





WTTTmTmmTmmmMmWA 



Fig. 60. 



possible sliding on each other. Spreading in layers has the 
advantage of partially ramming each layer, so that the subse- 
quent shrinkage is ver^^ small. Sometimes small trenches are 
dug along the lines of the toes of the embankment. This will 
frequently prevent the sliding of a large mass of the embank- 
ment, which will then require extensive and costly repairs, to 
say nothing of possible accidents if the sliding occurs after the 
road is in operation. Incidentally these trenches will be of 
value in draining the subsoil. When circumstances require an 
embankment on a hillside, it is advisable to cut out ''steps" to 
prevent a possible sliding of the whole embankment. Merely 
ploughing the side-hill will often be a cheaper and sufficiently 
effective method. 




Fig. 61. 

Occasionally the formation of a very high and long embank- 
ment may be most easily and cheaply accomplished by building 
a trestle to grade and opening the road- Earth can then be 



116 RAILROAD CONSTRUCTIOX. § 99. 

procured where most convenient, perhaps several miles away, 
loaded on cars with a steam-shovel, hauled by the trainload, and 
dumped from the cars with a patent imloader. On such a large 
scale, the cost per yard would be very much less than by ordi- 
nary methods — enough less sometimes to more than pay for the 
temporary trestle, besides allowing the road to be opened for 
traffic very much earlier, which is often a matter of prime 
financial importance. It may also obviate the necessity for 
extensive borrow-pits in the immediate neighborhood of the 
heavy fill and also utilize material which would otherwise be 
wasted. 

COMPUTATION OF HAUL. 

99. Nature of subject. As will be shown later when analyz- 
ing the cost of earthwork, the most variable item of cost is that 
depending on the distance hauled. As it is manifestly imprac- 
ticable to calculate the exact di^^tance to which every individual 
cartload of earth has been moved, it becomes necessary to devise 
a means which will give at least an equivalent of the haulage of 
all the earth moved. Evidently the average haul for any mass 
of earth moved is equal to the distance from the center of grav- 
ity of the excavation to the center of gravity of the embank- 
ment formed bv the excavated material. As a rough approxi- 
mation the center of gravity of a cut (or fill) may sometimes be 
considered to coincide with the center of gravity of that part of 
the profile repn^senting it, but the error is frequently very large. 
The c(inter of gravity may be determined by various methods, 
but the method of the ^' mass diagram'' accomplishes the same 
ultimate purpose (the determination of the haul) with all-suffi- 
cient accuracy and also furnishes other valual)le information. 

100. Mass diagram. In Fig. 62 let A/B' . . C/ represent 
a profile and grade Wmt drawn to the usual scales. Assume A' 
to be a pomt past which no earthwork will be hauled. Such 
a point is determined by natural conditions, as, for example, a 
river crossing, or one end of a long level stretch along which 
no grading is to be done except the formation of a low embank- 
ment from the material excavated from ample drainage ditches 
on each side. Above the profile draw an indefinite horizontal line 
{ACn in Fig. 62), which may Ije called the "zero line." Above 
every station point in the profile draw an ordinate (above or be- 



loO. 



EARTHV.'ORK. 



117 



low the zero line) which will represent the algebraic sum of 
the cubic yards of cut and fill 
(calling cut + and fill — ) from 
the point A^ to the point con- 
sidered. The computations of 
these ordinates should first be 
made in tabular form as shown 
below. In doing this shrinkage 
must be allowed for by consider- 
ing hoAv much embankment 
would actually be made by so 
many cubic yards of excavation 
of such material. For example, 
it will be found that 1000 cubic 
yards of sand or gravel, measured 
in place (see § 97), will make 
about 920 cubic yards of embank- 
ment; therefore all cuttings in 
sand or gravel should be dis- 
counted in about this propor- 
tion. Excavations in rock should 
be increased in the proper 
ratio. In short, all excavations 
should be valued according to the 
amount of settled embankment 
that could be made from them. 
Place in the first column a list 
of the stations; in the second 
column,the number of cubicyards 
of cut or fill between each station 
and the preceding station; in 
the third and fourth columns, the kind of material and the proper 
shrinkage factor; in the fifth column, a repetition of the quan- 
tities in cubic yards, except that the excavations are dinnnished 
(or increased, in the case of rock) to the number of cubic yards 
of settled embankment which may be made from them. In 
the sixth column place the algebraic sum of the quantities in the 
fifth column (calling cuts + and fills -) from the starting- 
point to the station considered. These algebraic sums at each 
station will be the ordinates, drawn to some scale, of the mass 
curve. The scale to be used will depend somewhat on whether 




118 



RAILROAD CONSTRUCTION. 



§101. 



the work is heavy or light, but for ordinary cases a scale of 
5000 cubic yards per inch may be used. Drawing these ordi- 
nates to scale, a curve A, B, . . . G may be obtained by joining 
the extremities of the ordinates. 



Sta. 


Yards] -/■! 


Material. 


Shrinkage 
factor. 


Yards, 

reduced 

for 

shrinkai^e. 


Ordinate 

in mass 

curve. 


46 + 70 













47 

48 

4- 60 
49 


+ 195 
4- 1792 
+ 614 

- 143 

- 906 

- 1985 

- 1721 

- 112 
+ 177 
4- 180 

- 52 

- 71 
4- 276 
4- 1242 
+ 1302 


Clayey soil 


- 10 per cent 

- 10 

- 10 


+ 175 
+ 1613 
+ 553 

- 143 

- 906 

- 1985 

- 1721 

- 112 
+ 283 
+ 289 

- 52 

- 71 
+ 249 
+ 1118 
+ 1172 


+ 175 

+ 1788 
4 2341 
4 2198 


50 






+ 1292 


51 






- 693 


52 






- 2414 


4- 30 






- 2526 


53 

+ 70 
54 


Hard rock 


4- 60 per cent 
+ 60 


- 2243 

- 1954 

- 2006 


4- 42 






- 2077 


55 
56 
57 


Clayey soil 


- 10 per cent 

- 10 
-- 10 


- 1828 

- 710 
+ 462 



loi. Properties of the mass curve. 

1. The curve will be rising while over cuts and falling while 
over fills. 

2. A tangent to the curve will be horizontal (as at B^ D, E, 
F, and G) when passing from cut to fill or from fill to cut. 

3 When the curve is helov: the "zero line" it shows that 
miaterial must be drawn backward (to the left) ; and vice versa, 
when the curve is above the zero line it shows that material 
must be drawn forward (to the right) . 

4. When the curve crosses the zero line (as at A and C) it 
show^s (in this instance) that the cut betAveen A' and B^ will just 
provide the material required for the fill between B^ and C\ and 
that no material should be hauled past C\ or, in general, past 
any intersection of the mass curve and the zero line. 

5. If any horizontal line be drawn (as ab), it indicates that 
the cut and fill between a' and ?;' will just balance. 

6. When the center of gravity of a given volume of material 
is to be moved a given distance, it makes no difference (at least 
theoretically) how far each individual load may be hauled or 
how any individual load may be disposed of. The summation 



§ 101. EARTHWORK. 119 

of the products of each load times the distance hauled will be a 
constant, whatever the method, and will equal the total volume 
times the movement of the center of gravity. The average 
haul, which is the movement of the center of gravity, will there- 
fore equal the summation of these products divided by the total 
volume. If we draw two horizontal parallel lines at an infini- 
tesimal distance dx apart, as at ab, the small increment of cut 
dx.Sit a' will fill the corresponding increment of fill at h^ , and 
this material must be hauled the distance ab.. Therefore the 
product of ab and dx, which is the product of distance times 
volume, is represented by the area of the infinitesimal rectangle 
at ab, and the total area ABC represents the summation of 
volume times distance for all the earth movement between A^ 
and C This summation of products divided by the total 
volume gives the average haul. 

7. The horizontal line, tangent at E and cutting the curve 
at e, f, and g, shows that the cut and fill between e^ and E' will 
just balance, and that a possible method of hauling (whether 
desirable or not) would be to " borrow" earth for the fill between 
C and c', use the material between D^ and £"' for the fill between 
e' and D', and similarly balance cut and fill between £" and /' 
and also between /' and g\ 

8. Similarly the horizontal line hklm may be dra^^-n cutting 
the curve, which will show another possible method of hauling. 
According to this plan, the fill between C and h^ would be 
made b}^ borrowing; the cut and fill between /z' and k' would 
balance; also that between A;' and /' and Ijetween /' and m\ 
Since the area eJiDkE represents the measure of haul for the 
earth between e' and E', and the other areas measure the corre- 
sponding hauls similarly, it is evident that the sum of the areas 
eJiDkE and ElFmf, which is the measure of haul of all the 
material between e' and /', is largely in excess of the sum of 
the areas hDk, kEl, and IFm, plus the somewhat uncertain 
measures of haul due to borrowing material for e/h' and \^asting 
the material between m' and /'. Therefore to make the meas- 
ure of haul a minimum a line should be dravrn which will make 
the sum of the areas between it and the mass curve a minimum. 
Of course this is not necessarily the cheapeit plan, as it implies 
more or less borrowing and wasting of material, which may 
cost more than the amount saved in haul. The compd.rison of 
the two methods is quite simple, however. Since the amount 



120 RAILROAD CONSTRUCTION. § 102. 

of fill between e' and h' is represented by the difference of the 
ordinates at e and h, and similarly for m' and /', it follows that 
the amount to be borrowed between e' and h' will exactly equal 
the amount wasted between m' and /'. By the first of the above 
methods the haul is excessive, but is definitely known from the 
mass diagram, and all of the material is utihzed; by the second 
method the haul is reduced to about one-half, but there is a 
known quantity in cubic yards wasted at one place and the same 
quantity borrowed at another. The length of haul necessary 
for the borrowed material would need to be ascertained; also 
the haul necessary to waste the other material at a place where 
it would be unobjectionable. Frequently this is best done by 
widening an embankment beyond its necessaiv width. The 
computation of the relative cost of the above methods will be 
discussed later (§ 116). 

9. Suppose that it were deemed best, after drawing the mass 
curve, to introduce a trestle between s' and i'', thus saving an 
amount in fill equal to tv. If such had been the original design, 
the mass curv^e would have been a straight horizontal line between 
s and t and \A'ould contmue as a curve which would be at all 
points a distance tv above the curve vFmzfGg, If the line Ef is 
to be used as a zero line, its intersection with the new curve at x 
will show that the material between E^ and z^ will just balance 
if the trestle is used, and that the amount of haul will be meas- 
ured by the area between the line Ex and the broken line Estx. 
The same computed result may be obtained without drawing 
the auxiliary curve txn ... by drawing the horizontal line zy 
at a distance xz{=tv) below Ex. The amount of the haul can 
then be obtained by adding the triangular area between Es and 
the horizontal line Ex, the rectangle between st and Ex, and the 
irregular area between vFz and y . . . z (which last is evidently 
equal to the area between tx and E . . . x). The disposal of the 
material at the right of z' would then be governed by the nidica- 
tions of the profile and mass diagram which would be found at 
the right of cj' . In fact it is difficult to decide with the best of 
judgment as to the proper disposal of material without having 
a mass diagram extending to a considerable distance each side 
of that part of the road under immediate consideration. 

102. Area of the mass curve. The area may .be computed 
most readily by means of a planimeter, which is capable with 
reasonable care of measuring such areas with as great accuracy 



§ 103. EARTH^.VORK. 121 

as is necessary for this T\'ork. If no such instrument is obtain- 
able, the area may be obtained by an application of '' Simpson's 
rule." The ordinates will usually be spaced 100 feet apart. 
Select an even number of such spaces, leaving, if necessary, one 
or more triangles or trapezoids at the ends for separate and 
independent computation. Let j/q . . . yn be the ordinates, i.e., 
the number of cubic 3^ards at each station of the mass curve, or 
the figures of ^'column six" referred to in § 100. Let the uni- 
form distance between ordinates (--=100 feet) be called 1, i.e., 
one station. Then the units of the resulting area will be cubic 
yards hauled one station. Then the 

Area = iL2/o 4- 4(2/1 + 2/3-^ • • .2/(n-l) + 2(2/2 + 2/4+ • . .2/(»t-2)+2/J- (70) 

When an ordinate occurs at a substation, the best plan is to 
ignore it at first and calculate the area as above. Then, if the 
difference involved is too great to be neglected, calculate the 
area of the triangle having the extremity of the ordinate at the 
substation as an apex, and the extremities of the ordinates at the 
adjacent stations as the ends of the base. This may be done by 
finding the ordinate at the substation that would be a propor- 
tional between the ordinates at the adjacent full stations. Sub- 
tract this from the real ordinate (or vice versa) and multiph^ the 
difference by JXl. An inspection will often show that the 
correction thus obtained would be too small to be worthy of con- 
sideration. If there is more than one substation between two 
full stations, the corrective area will consist of two triangles and 
one or more trapezoids which may be similarly computed, if 
necessary. 

When the zero line (Fig. 62) is shifted to eE, the drop from 
AC (produced) to E is known in the same units, cubic yards. 
This constant may be subtracted from the numbers ('^column 
6," § 100) representing the ordinates, and will thus give, with- 
out any scaling from the diagram, the exact value of the modi- 
fied ordinates. 

103. Value of the mass diagram. The great value of the mass 
diagram lies in the readiness with which different plans for the 
disposal of material may be examined and compared. When 
the mass curve is once drawn, it will generally require only a 
shifting of the horizontal line to show the disposal of the material 
by any proposed method. The mass diagram also shows the 



122 RAILROAD CONSTRUCTION. § 104. 

extreme length of haul that will be required by any proposed 
method of disposal of material. This brings into consideration 
the ^' limit of profitable haul/' which will be fully discussed in 
§ 116. For the present it may be said that with each method 
of carrying material there is some limit beyond which the expense 
of hauling w^ill exceed the loss resulting from borrowing and 
wasting. With wheelbarrows and scrapers the limit of profit- 
able haul is comparatively short, with carts and tram-cars it is 
much longer, while with locomotives and cars it may be several 
miles. If, in Fig. 62, eE or Ef exceeds the limit of profitable 
haul, it shows at once that some such line as hktm should be 
drawn and the material disposed of accordingly. 

104. Changing the grade line. The formation of the mass 
curve and the resulting plans as to the disposal of material are 
based on the mutual relations of the grade line and the surface 
profile and the amounts of cut and fill w^hich are thereby im- 
plied. If the grade line is altered, every cross-section is altered, 
the amount of cut and fill is altered, and the mass curve is also 
changed. At the farther limit of the actual change of the grade 
line the rcA'ised mass curve will have (in general) a different 
ordinate from the previous ordinate at that point. From that 
point on, the revised mass curve will be parallel to its former 
position, and the revised curve may be treated similarly to the 
case previously mentioned in which a trestle was introduced. 
Since it involves tedious calculations to determine accurately 
how much the volume of earthwork is altered by a change in 
grade line, especially through irregular country, the effect on 
the mass curve of a change in the grade line cannot therefore 
be readily determined except in an approximate way. Raising 
the grade line w^ill evidently increase the fills and diminish the 
cuts, and vice versa. Therefore if the mass curve indicated, for 
example, either an excessively long haul or the necessity for 
borrowing material (implying a fill) and wasting material 
farther on (implying a cut), it would be possible to diminish the 
fill (and hence the amount of material to be borrowed) by lower- 
ing the grade line near that place, and diminish the cut (and 
hence the amount of material to be wasted) by raising the 
grade line at or near the place farther on. Whether the advan- 
tage thus gained would compensate for the possibly injurious 
effect of these changes on the grade line would require patient 
investigation. But the method outlined shows how the mass 



§105. 



EARTHWORK, 



123 



curve might be used to indicate a possible change in grade line 
which might be demonstrated to be profitable. 

105. Limit of free haul. It is sometimes specified in con- 
tracts for earthwork that all material shall be entitled to free 
haul up to some specified limit, say 500 or 1000 feet, and that 
all material drawn farther than that shall be entitled to an 
allowance on the excess of distance. It is manifestly imprac- 
ticable to measure the excess for each load, as much so as to 
measure the actual haul of each load The mass diagram also 
solves this problem very readily. Let Fig. 63 represent a pro- 




FiG. 63. 



file and mass diagram of about 2000 feet of road, and siii>po«S& 
that 800 feet is taken a^ the limit of free haul. Find tw^o p6int*i, 
a and b, in the mass curve ivhich are on the same horizontal line 
and w^hich are 800 feet apart. Project these points down to a' 
and h'. Then the cut and fiU betw^een a' and h' will just balance, 
and the cut between A' and a' will be needed for the fill between 
6' and C\ In the mass curve, the area between the horizontal 
line ah and the curve aBb represents the haulage of the material 
between a' and b\ which is all free. The rectangle abmn repre- 
sents the haulage of the material in the cut A/ a' across the 800 
feet from a' to b\ This is also free. The sum of the two areas 
Aam and bnC represents the haulage entitled to an allowance, 
since it is the summation of the products of cubic yards times 
the excess of distanc^^ hauled. 

If the amount of cut and fill was symmetrical about the point 



124 RAILROAD CONSTRUCTION. § 105. 

B', the mass curve would be a syrrimetrical curve about the 
vertical line through B, and the two limiting h'nes of free haul 
would be placed symmetricall}^ about B and B\ In general 
there is no such synimetry, and frequently the difference is con- 
siderable The area aBbnm will be materially changed accord- 
ing as the two vertical hues a77i and bn^ always 800 feet apart, 
are shifted to the right or left. It is easy to show that the area 
aBbnm is a maximum when ab is horizontal. The minimum 
value Avould be obtained either when m reached A or n reached 
C, depending on the exact form of the curve. Since the posi- 
tion for the minimum value is manifestly unfair, the best definite 
value obtainable is the maximum, which nmst be obtained as 
above described. Since aBbnm is made maximum, the remainder 
of the area, which is the allowance for overhaul, becomes a mini- 
mum. The areas Aam and bCn may be obtained as in § 102. 
If the whole area AaBbCA has been previously computed, it 
may be more convenient to compute the area aBbnm and sub- 
tract it from the total area. 

Since the intersections of the mass curve and the ^'zero line" 
mark limits past which no material is drawn, it follows that 
there will be no allowance for overhaul except where the dis- 
tance between consecutive intersections of the zero line and mass 
curve exceeds the limit of free haul. 

Frequently all allowances for overhaul are disregarded: the 
profiles, estimates of quantities, and the required disposal of 
material are shown to bidding contractors, and they must then 
^make their own allowances and bid accordingly. This method 
has the advantage of avoiding possible disputes as to the amount 
of the overhaul allowance, and is popular with railroad com- 
panies on this account. On the other hand the facility with 
which different plans for the disposal of material may be studied 
and compared by the mass-curve method facilitates the adoption 
of the most economical plan, and the elimination of uncertainty 
will frequently lead to a safe reduction of the bid, and so the 
method is valuable to both the railroad company and the con- 
tractor. 

ELEMENTS OF THE COST OF EARTHWORK, 

(The following analysis of the cost of earthwork follows the 
general method given in the well-known papers published by 



§ 106. EARTHWORK. 125 

Ellwood Morris, C.E., in the Journal of the Franklin Institute 
in September and October, 1841. Numerous corroborative 
data have been obtained from various other sources, and also 
figures on methods not then in vogue.) 

io6. General divisions of the subject. The variations in the 
cost of earthwork are caused by the greatly varjdng conditions 
under which the work is done, chief among which is character 
of material, method of carriage, and length of haul. Any gen- 
eral system of computation must therefore differentiate the total 
cost into such elementary items that all differences due to varia- 
tions in conditions may be allowed for. The variations due to 
character of material Tsill be allowed for by an estimate on looc: 
light sandy soil, and also an estimate on the heaviest soils, sue.: 
as stiff clay and hard-pan. These represent the extremes (ex- 
cluding rock, which will be treated separately), and the cost o: 
intermediate ^ades must be estimated by interpolating between 
the extreme values. The general divisions of the subject will be : * 

1. Loosening. 

2. Loading. 

3. Hauling. 

4. Spreading. 

5. Keeping roadwa3^s in order. 

6. Repairs, wear, depreciation, and interest on cost of plant. 

7. Superintendence and incidentals. 

8. Contractor's profit. 

By making the estimates on the basis of $1 per day for the 
cost of common labor, it is a simple matter to revise the esti- 
mates according to the local price of labor by multiplying the 
final estimates of cost by the price of labor in dollars per day. 

107. Item I. Loosening, (a) Ploughs. Very light sandy 
soils can frequently be shovelled mthout any previous loosening, 
but it is generally economical, even with very light material, to 
use a plough. Morris quotes, as the results of experiments, 
that a three-horse plough would loosen from 250 to 800 cubic 
yards of earth per day, which at a valuation of $5 per day would 
make the cost per yard vary from 2 cents to 0.6 cent. Traut- 
wine estimates the cost on the basis of two men handling a two- 
horse plough at a total cost of $3.87 per day, being $1 each for 

* Trautwine. 



126 RAILROAD CONSTRUCTION. § lOS. 

the men, 75 c. /or each horse, and an allowance of 37 c. for the 
plough, harness, etc. From 200 to 600 cubic yards is estimated 
as a fair day's work, w^hich makes a cost of 1.9 c. to 0.65 c. per 
yard, which is substantially the same estimate as above. Ex- 
tremely heavy soils have sometimes been loosened by means of 
special ploughs operated by traction-engines. 

(b) Picks. When picks are used for loosening the earth, as 
is frequently necessary and as is often done when ploughing 
would perhaps be really cheaper, an estimate * for a fair dav's 
work is from 14 to 60 cubic yards, the 14 yards being the esti- 
mate for stiff clay or cemented gravel, and the 60 yards the esti- 
mate for the lightest soil that would require loosening. At $1 
per day this means about 7 c. to 1.7 c. per cubic yard, which is 
about three times the cost of ploughing. Five feet of the face 
is estimated t as the least width along the face of a bank that 
should be allowed to enable each laborer to worl* w^ith freedom 
and hence economically. 

(c) Blasting. Although some of the softer shaly rocks may 
be loosened with a pick for about 15 to 20 c. per yard, yet rock 
in general, frozen earth, and sometimes even compact clay are 
most economically loosened by blasting. The subject of blast- 
ing will be taken up later, §§ 117-123. 

(d) Steam-shovels. The items of loosening and loading 
merge together wdth this method, which will therefore be treated 
in the next section. 

io8. Item 2. Loading, (a) Hand-shovelling. Much depends 
on proper management, so that the shovellers need not wait un- 
duly either for material or carts. With the best of managem.ent 
considerable time is thus lost, and yet the intervals of rest 
need not be considered as entirely lost, as it enables the men to 
work, while actually loading, at a rate w^hich it would be physi- 
cally impossible for them to maintain for ten hours. Seven 
shovellers are sometimes allowed for each cart; otherwise there 
should be five, two on each side and one in the rear. Economy 
requires that the number of loads per cart per day should be 
made as large as possible, and it is therefore wise to employ as 
many shovellers as can work without mutual interference and 
without wasting time in w^aiting for material or carts. The 
figures obtainable for the cost of this item are unsatisfactory on 

* Trautwine. t Hurst. 



§ 108. EARTHWORK. 127 

account of their large disagreements. The following are quoted 
as the number of cubic yards that can be loaded into a cart by 
an av^erage laborer in a working day of ten hours, the lower 
estimate referring to heavy soils, and the higher to light sandy 
soils: 10 to 14 cubic yards (Morris), 12 to 17 cubic yards (Has- 
koU), 18 to 22 cubic yards (Hurst), 17 to 24 cubic yards (Traut- 
wine), 16 to 48 cubic 3'ards (Ancelin). As these estimates are 
generally claimed to be based on actual experience, the discrep- 
ancies are probably due to differences of management. If the 
average of 15 to 25 cubic yards be accepted, it means, on the 
basis of $1 per day, 6.7 c. to 4 c per cubic yard. These esti- 
mates apply only to earth. Rockwork costs more, not only 
because it is harder to handle, but because a cubic yard of solid 
rock, measured in place, occupies about 1.8 cubic yards when 
broken up, while a cubic yard of earth will occupy about 1.2 
cubic yards. Rockwork will therefore require about 50% more 
loads to haul a given volume, measured in place, than will the 
same nominal volume of earthwork. The above authorities give 
estimates for loading rock varying from 6.9 c. to 10 c. per cubic 
yard. The above estimates apply only to the loading of carts 
or cars with shovels or by hand (loading masses of rock). The 
cost of loading wheelbarrows and the cost of scraper w^ork will 
be treated under the item of hauling. 

(b) Steam-shovels.* Whenever the magnitude of the w^ork 
will Avarrant it there is great economy in the use of steam-shovels. 
These have a ^'bucket" or ^'dipper" on the end of a long beam, 
the bucket having a capacity A^arying from | to 2J cubic yards. 
Steam-shovels handle all kinds of material from the softest 
earth to shale rock, earthy material containing large boulders, 
tree-stumps, etc. The capacity of the larger sizes is about 3000 
cubic yards in 10 hours. They perform all the work of loosen- 
ing and loading. Their economical working requires that the 
material shall be hauled away as fast as it can be loaded, 
Avhich usually means that cars on a track, hauled by horses or 
mules, or still better by a locomotive, shall be used. The ex- 
penses for a steam-shovel, costing about $5000, will average 
about $1000 per month. Of this the engineer will get $100; the 

* For a thorough treatment of the capabilities, cost, and management 
of steam-shovels the reader is referred to " Steam-shovels and Steam-shovel 
Work," by E. A. Hermann. D. Van Nostraad Co., New York. 



128 RAILROAD CONSTRUCTION. § 109. 

fireman $50 ; the cranesman $90 ; repafe perhaps $250 to $300 ; 
coal, from 15 to 25 tons, cost very variable on account of expen- 
sive hauling ; water, a very uncertain amount, sometimes costing 
$100 per month; about five laborers and a foreman, the laborers 
getting $1.25 per day and the foreman $2.50 per day, which will 
amount to $227.50 per month. This gang of laborers is em- 
ployed in shifting the shovel when necessary, taking up and 
relaying tracks for the cars, shifting loaded and unloaded cars, 
etc. In shovelling through a deep cut, the shovel is operated 
so as to undermine the upper parts of the cut, which then fall 
down within reach of the shovel, thus increasing the amount 
of material handled for each new position of the shovel. If the 
material is too tough to fall down by its own weight, it is some- 
times found economical to employ a gang of men to loosen it or 
even blast it rather than shift the shovel so frequently. Non- 
condensing engines of 50 horse-power use so much v/ater that 
the cost of water-supply becomes a serious matter if water is 
not readily obtainable. The lack of water facilities will often 
justify the construction of a pipe line from some distant source 
and the installation of a steam-pump. Hence the seemingly 
large estimate of $100 per month for water-supply, although 
under favorable circumstances the cost may almost vanish. 
The larger steam-shovels will consume nearly a ton of coal per 
day of 10 hours. The expense of hauling this coal from the 
nearest railroad or canal to the location of the cut is often a very 
serious item of expense and may easily double the cost per ton. 
Some steam -shovels have been constructed to be operated by 
electricity obtained from a plant perhaps several miles away. 
Such a method is especially advantageous when fuel and water 
are difficult to obtain. 

109. Item 3. Hauling. The cost of hauling depends on the 
number of round trips per day that can be made b}' each vehicle 
employed. As the cost of each vehicle is practically the same 
whether it makes many trips or few, it becomes important that 
the number of trips should be made a maximum, and to that 
end there should be as little delay as possible in loading and un- 
loading. Therefore devices for facilitating the passage of the 
vehicles have a real money value. 

(a) Carts. The average speed of a horse hauling a two- 
wheeled cart has been found to be 200 feet per minute, a little 
slower when hauling the load and a little faster when returning 



§ 109. EARTHWORK. 129 

empty. This figure has been repeatedly verified. It means an 
allowance of one minute for each 100 feet (or " station '') of 
''lead — the lead being the distance the earth is hauled." The 
time lost in loading, dumping, waiting to load, etc., has been 
found to average 4 minutes per load. Representing the num- 
ber of stations (100 feet) of lead by s, the number of loads 
handled in 10 hours (600 minutes) would be 600 -^(s + 4). The 
number of loads per cubic yaif., measured in the bank, is differ- 
entiated by Morris into three classes, viz. : 

3 loads per cubic yard in descending hauling; 
Si " " '' " " level hauling; and 
4: " '' " '' '' ascending hauling. 

Attempts have been made to estimate the effect of the grade 
of the roadway by a theoretical consideration of its rate, and of 
the comparative strength of a horse on a le^xl and on various 
grades. While such computations are alw^ays practicable on a 
railway (even on a temporary construction track), the traction 
on a temporary earth roadway is always very large and so very 
variable that any refinements are useless. On railroad earth- 
work the hauling is generally nearly level or it is descending — 
forming embankments on low ground with material from cuts in 
high ground. The onl}^ common exception occurs when an 
embankm.ent is formed from borrow-pits on low ground. One 
method of allowing for ascending grade is to add to the hori- 
zontal distance 14 times the difference of elevation for work 
with carts and 24 times the difference of elevation for work 
with wheelbarrows, and use that as the lead. For example, 
using carts, if the lead is 300 feet and there is a difference of 
elevation of 20 feet, the lead would be considered equivalent to 
300 + (14X20) =580 feet on a level. 

Trautwine assumes the average load for all classes of work 
to be J cubic yard, which figure is justified by large experience. 
Using one figure for all classes of work simplifies the calculations 
and gives the number of cubic yards carried per day of 10 hours 

equal to — — — . Dividing the cost of a cart per day by the 

number of cubic yards carried gives the cost of hauling per 
yard. In computing the cost of a cart per day, Trautwine 
refers to the practice of having one driver manage four carts, 
thus making a charge of 25 c. per day for each cart for the driver. 



130 RAILROAD CONSTRUCTION. § 109. 

75 c. is allowed for the horse, which is supposed to be the total 
cost, including that for Sundays and rainy days. 25 c. more is 
allowed for the cart, harness, repairs, etc., thus making a total 
cost of $1.25 per day. Some contractors employ a greater num- 
ber of drivers and expect each to assist in loading. There is 
found to be no saving in total cost per yard, while the chances 
of loafing are perhaps greater. Morris instances five actual cases 
in which the cost of the cart (reduced to the basis of $1 per day 
for labor) varied from $1.37 to $1.48. The items of these costs 
were not given. 

Since the time required for loading loose rock is greater than 
for earthwork, less loads will be hauled per day. The tim.e 
allowance for loading, etc., is estimated by Trautwine as 6 
minutes instead of 4 as for earth. Considering the great ex- 
pansion of rock when broken up (see § 97), one cubic yard of 
solid rock, measured in place, would furnish the equivalent of 
five loads of earthwork of J cubic yard. Therefore, on the 
basis of five loads per cubic yard, the number of cubic yards 

handled per day per cart would be -7 -rr. 

^ -^ ^ 5(s + 6) 

^ . ,. . 125X5(s + 6) _,, 

Cost per yard m cents = ^^ -. . . (71) 

(b) Wagons. For longer leads (i.e., from J to § of a mile) 
wagons drawn by two horses have been found most economical. 
The w^agons have bottoms of loose thick narrow boards and are 
unloaded very easily and quickly by lifting the individual boards 
and breaking up the continuity of the bottom, thus depositing 
the load directly underneath the wagon. The capacity is about 
one cubic yard. The cost may be estimated on the same prin- 
ciples as that for carts. 

(c) Wheelbarrows. According to Trautwine, the speed of 
moving w^heelbarrows may be considered the same as for carts, 
200 feet per minute; the time spent in loading and dumping is 
li minutes, and in addition about j\ of the time is wasted in 
short rests, adjusting the wheeling planks, etc. On the basis of 
$1 per day for labor, an allowance of 5 c. for the barrow, and 14 
loads per cubic yard, the cost of hauling per cubic yard (com- 
puted on the same principles as above) will be 

105X14(8 + 1.25) 



600X0.9 



(72) 



§ 109. EARTHWORK. 131 

For rockwork the number of loads per cubic yard is estimated 
as 24, and the time spent in loading, etc., estimated at 1.6 min- 
; ,utes instead of 1.25 minutes, which makes the estimate 

n . I • A lQ5X24(s + 1.6) ,_^ 

Cost per cubic yard = ^q^^q^ . . . (73) 

(d) Scrapers.* Scrapers, or scoops, are especially useful in 
canal work, and also for railroad work when a low embankment 
is to be formed from borrow-pits at the sides, when the distance 
does not exceed 100 feet, nor the vertical height 15 feet. The 
slope should not exceed 1.5 to 1. Under these conditions scraper 
work is cheaper than any other method. Scooping may be done 
all in one direction, in which case two half -turns are made for 
each load moved; or it may be done in both directions (from 
both sides on to a bank, or, in canal work, from the center to 
each bank), in which case one load is hauled to each haLf-tum. 
The capacity of the scoops (the ''drag" variety) is ^V cubic 
yard; the time lost in loading, unloading, and all other ways 
per load (except in turning) will average f minute ; the time lost 
in each half -turn (semi-circle) is J minute; the speed of the 
horses may be estimated as 70 feet of lead per minute, the lead 
being here considered as the sum of the vertical and horizontal 
distances, and the estimate including the time of going and re- 
turning. If a represents the sum of the horizontal and vertical 
distances, the number of cubic yards handled per day of 10 
hours by ''side-scooping" will be 

''' ■ 4200 



O.ll a ,1,1, which equals 



For "double-scooping" the formula becomes 

^^^ • . 4200 

, which equals ^ . 

Dividing the cost of a scraper per day (estimated at $2.75) by 
the number of yards handled per day gives the average cost per 
yard. 




* Condensed from Journ. Franklin Inst., Oct. 1841, by Morri?. 



132 RAILROAD CONSTRUCTION. § 109. 

Except in very loose sandy soil it is best to plough the earth 
first, which will cost about 1 c. per yard. (See § 107.) Drag- 
scrapers are no\A' made chiefly of steel, and their capacity is more 
nearly 0.15 cubic yard. Wheeled scrapers, having a capacity 
of about 0.5 cubic yard, are frequently used with even greater 
economy and for greater distances, as they are cheaper than 
carts up to 250 or 300 feet of lead. Both drag- and wheel- 
scrapers are best operated in gangs of perhaps 10, using extra 
or ^'snap" teams to help load, and a few extra men to help in 
loading and unloading. The average cost of one scraper per 
day may thus be easily calculated and the average number of 
cubic yards handled per day computed as above, from which 
the cost per j^ard may be estimated. 

(e) Cars and horses. The items of cost by this method are 
(a) charge for horses employed, (h) charge for men employed 
strictly in hauling, (c) charge for shifting rails when necessary, 
(d) repairs, depreciation, and interest on cost of cars and track. 
Part of this cost should strictly be classified under items 5 and 
6, mentioned in § 106, but it is perhaps more convenient to 
estimate them as follows: 

The traction of a car on rails is so very small and constant 
that grade resistance constitutes a very large part of the tolal 
resistance if the grade is 1% or more. For a 1 ordinary grades 
it is sufficiently accurate to say that the grade resistance is to 
the gross weight as the rise is to the distance. If the distance 
is supposed to be measured along the slope, the proportion is 
strictly true; i.e., on a 1% grade the grade resistance is 1 lb. 
per 100 of weight or 20 lbs. per ton. If the resistance on a 
level at the usual velocity is j^q, a grade of 1 : 120 (0.83%) will 
exactl}^ double it. If the material is hauled down a grade of 
1 : 120, the cars will run by gravity after being started. The 
work of hauling will then consist practically of hauling the 
empty cars up the grade. The grade resistance depends only 
on the rate of grade and the weight, but the tractive resistance 
will be greater per ton of weight for the unloaded than for the 
loaded cars. The tractive power of a horse is less on a grade 
than on a level, not only because the horse raises his own weight 
in addition to the load, but is anatomically less capable of 
pulling on a grade than on a level. In general it will be pos- 
sible to plan the work so that loaded cars need not be hauled up 
a grade, unless an embankment is to be formed from a low 



§ 109. EARTHWORK. 133 

boiTow-pit, in which case another method would probably be 
advisable. These computations are chiefl}^ utilized in designing 
the method of work — the proportion of horses to cars. An 
example may be quoted from English practice (Hurst), in which 
the cars had a capacity of 3J cubic yards, weighing 30 cwt. 
empty. Two horses took five ^'wagons" f of a mile on a level 
railroad and made 15 journeys per day of 10 hours, i.e., they 
handled 250 yards per day. In addition to those on the 
''straight road/' another horse was employed to make up the 
train of loaded wagons. With a short lead the straight -road 
horses were employed for this purpose. In the aboA^e example 
the number of men required to handle these cars, shift the 
tracks, etc., is not given, and so the exact cost of the above 
work cannot be analyzed. It may be noticed that the two 
horses travelled 22 J miles per day, drawing in one direction a 
load, including the weight of the cars, of about 57,300 lbs., or 
28.65 net tons. Allowing ^^^ as the necessary tractive force, 
it would require a pull of 477.5 lbs., or 239 lbs. for each horse. 
With a velocity of 220 feet per minute this would amount to 
IJ horse-power per horse, exerted for only a short time, how- 
ever, and allowing considerable time for rest and for drawing 
only the empty cars. The cars generally used in this countiy 
have a capacity" of IJ cubic yards and cost about $65 apiece. 
Besides the shovellers and dumping-gang, several men and a 
foreman will be required to keep the track in order and to make 
the constant shifts that are necessary. Two trains are generally 
used, one of which is loaded while the other is run to the dump. 
Some passing-place is necessary, but this is generally provided 
by having a switch at the cut and running the trains on each 
track alternately. This insures a train of cars always at the cut 
to keep the shovellers emplo}'ed. The cost of hauling per cubic 
yard can only be computed when the number of laborers, cars, 
and horses employed are known, and these will depend on the 
lead, on the character of the excavation, on the grade, if any, 
etc., and must be so proportioned that the shovellers need not 
wait for cars to fill, nor the dumping-gang for material to handle, 
nor the horses and drivers for cars to haul. ]\Iuch skill is neces- 
sary to keep a large force in smooth running order. 

(f) Cars and locomotives. 30-lb. rails are the lightest that 
should be used for this work, and 35- or 40-lb. rails are better. 
One or two narrow-gauge locomoii^es (depending on the length 



134 RAILROAD CONSTRUCTION. § HO. 

of haul), costing about $2500 each, will be necessary to handle 
two trains of about 15 cars each, the cars having a capacity of 
about 2 cubic yards and costing about $100 each. Some cars 
can be obtained as low as $70. A force of about five mea nnd 
a foreman will be required to shift the tracks. The track- 
shifters, except the foreman, may be common laborers. The 
dumping-gang will require about seven men. Even when the 
material is all taken down grade the grades may be too steep for 
the safe hauling of loaded cars do^^n the grade, or for hauling 
empty cars up the grade. Under such circumstances temporary 
trestles are necessary to reduce the grade. When these are 
used, the uprights and bracing are left in the embankment- 
only 'the stringers being removed. This is largely a necessity, 
but is partially compensated by the fact that the trestle forms a 
core to the embankment which prevents lateral shifting during 
settlement. The average speed of the trains may be taken as 
10 miles per hour or 5 miles of lead per hour. The time lost 
in loading and unloading is estimated (Trautwine) as 9 minutes 
or .15 of an hour. The number of trips per day of 10 hours 

10 50 ^^^ I 

will equal f^i^i^^^n^^d)T7l5 '''* (miles of lead) + .75' 
course this quotient must be a whole number. Knowing the 
number of trains and their capacity, the total number of cubic 
yards handled is known, which, divided into the total daily cost 
of the trains, will give the cost of hauling per yard. 1 he daily 
cost of a train will include 

(a) Wages of engineer, who frequently fires his o^^'n engine; 

(b) Fuel, about i to 1 ton of bituminous coal, depending on 

work done; 

(c) Water, a very variable item, frequently costing $3 to $5 

(d) Repairs, variable, frequently at rate of 50 to 60% per ' 

year ; 

(e) Interest on cost and depreciation, 16 to 40^: . 
To these must be added, to obtain the total cost of the haul, 
(/) Wages of the gang employed in shifting track, 
no. Choice of method of haul dependent on distance. In 

light side-hill work in which material need not be mo^'ed more 
than 12 or 15 feet, i.e., moved laterally across the roadbed, the 
earth mav be moved most cheaply by mere shoveUing. Beyond 
12 feet scrapers are more economical. .At about 100 feet drag- 



§ 111. EARTHWORK. 135 

scrapers and wheelbarrows are equally economical. Between 
100 and 200 feet wheelbarrows are generally cheaper than either 
carts or drag-scrapers, but ^\ heeled scrapers are always cheaper 
than wheel})arrows. Beyond 500 feet two-wheeled carts become 
the most economical up to about 1700 feet; then four-wheeled 
wagons become more economical up to 3500 feet. Beyond this 
cars on rails, drawn by horses or by locomoti^■es, become cheaper. 
The economy of cars on rails becomes evident for distances as 
small as 300 feet provided the volume of the excavation will 
justify the outlay. Locomotives will always be cheaper than 
horses and mules providing the work to be done is of sufficient 
magnitude to justify the purchase of the necessary plant and 
risk the loss in selling the plant ultimately as second-hand equip- 
ment, or keeping the plant on hand and idle for an indefinite 
period waiting for other work. Horses will not be economical 
for distances much over a mile. For greater distances locomo- 
tives are more economical, but the question of ^' limit of profit- 
able haul" (§ 116) must be closely studied, as the circumstances 
are certainly not common when it is advisable to haul material 
much over a mile. 

III. Item 4. Spreading. The cost of spreading varies with 
the method employed in dumping the load. When the earth is 
tipped over the edge of an embankment there is little if any 
necessary work. Trautwine allows about \ c. per cubic yard 
for keeping the dumping-places clear and in order. This would 
represent the wages of one man at $1 per day attending to the 
unloading of 1200 t^^ o-wheeled carts each carr^-ing J cubic yard. 
1200 carts in 10 hours would mean an average of two per minute, 
which implies more rapid and efficient work than may be de- 
pended on. The allowance is probably too small. AVhen the 
material is dumped in layers some levelling is required, for which 
Traut^^•ine allows 50 to 100 cubic yards as a fair day's work, 
costing from 1 to 2 cents per cubic yard. The cost of spread- 
ing will not ordinarily exceed this and is frequently nothing — 
all depending on the method of unloading. It should be noted 
that Mr. Morris's examples and computations (Jour. Franklin 
Inst., Sept. 1841) disregard altogether any special charge for 
this item. 

j 112. Item 5. Keeping Roadways in order. This feature 
is important as a measure of true economy, whatever the s}'stem 
of transportation, but it is often neglected. A pett}^ saving in 



136 RAILROAD CONSTRUCTION. § 113. 

such matters will cost many times as much in increased labor in 
hauling and loss of time. With some methods of haul the cost 
is best combined with that of other items. 

(a) Wheelbarrows. Wheelbarrows should generally be run 
on planks laid on the ground. The adjusting and shifting of these 
planks is done by the wheelers, and the time for it is allowed for 
in the 10% allowance for '^ short rests, adjusting the wheeling 
plank, etc.'^ The actual cost of the planks must be added, but 
it would evidently be a very small addition per cubic yard in a 
large contract. When the Avheelbarrows are run on planks 
placed on ^'horses'' or on trestles the cost is very appreciable; 
but the method is frequently used with great economy. The 
variations in the requirements render any general estimate of 
such cost impracticable. 

(b) Carts and wagons. The cost of keeping roadways in order 
for carts and wagons is sometimes estimated merely as so much 
per cubic yard, but it is evidently a function of the lead. The 
work consists in draining off puddles, filling up ruts, picking 
up loose stones that may have fallen off the loads, and in general 
doing everything that will reduce the traction as much as possi- 
ble. Temporary inclines, built to avoid excessive grade at some 
one point, are often measures of true economy. Trautwine 
suggests yV ^' P^^ cubic yard per 100 feet of lead for earthwork 
and 1^0 c. for rockwork, as an estimate for this item when carts 
are used. 

(c) Cars. When cars are used a shifting-gang, consisting of 
a foreman and several men (say five), are constantly employed 
in shifting the track so that the material may be loaded and un- 
loaded where it is desired. The average cost of this item may 
be estimated by dividing the total daily cost of this gang by the 
number of cubic yards handled in one day. 

113. Item 6. Repairs, Wear, Depreciation, and Interest 
ON COST OF Plant. The amount of this item evidently depends 
upon the character of the soil — the harder the soil the worse the 
wear and depreciation. The interest on cost depends on the 
current borrowing value of money. The estimate for this item 
has already been included in the allowances for horses, carts, 
ploughs, harness, wheelbarrows, steam-shovels, etc. Trautwine 
estimates \ c. per cubic yard for picks and shovels. Deprecia- 
tion is generally a large percentage of the cost of earth-working 



§ 114. EARTHWORK. 137 

tools, the life of all being limited to a few years, and of many 
tools to a few months. 

114. Item 7. SUPERINTENDENCE AND INCIDENTAL'S. The inci- 
dentals include water-carriers, trimmmg cuts to grade^ digging 
the side ditches, trimming up the sides of borrow-pits to prevent 
their becoming unsightly^ etc. These last operations yield but 
little earth and cost far more than the price paid per cubic yard. 
Morris allows 1 c. per cubic yard for this item; Trautwine allows 
IJ to 2 c. for it; while others combine items 6 and 7 and call 
them 5% of the total cost, which method has the merit of mak- 
ing the cost of items 6 and 7 a function of the character of soil 
and length of lead. 

115. Item 8. CONTRACTOR'S PROFIT. This is usually esti- 
mated at from 6 to 15%, according to the sharpness of the com- 
petition and the possible tmcertainty as to true cost owing to 
unfavorable circumstances. The contractor's real profit may 
vary considerably from this. He often pays clerks^ boards and 
lodges the laborers in shanties built for the purpose, or keeps a 
supply-store, and has various other items both of profit and 
expense. His profit is largely dependent on skill in so handling 
the men that all can work effectively without interference or 
delays in waiting for others. An unusual season of bad weather 
will often affect the cost very seriously. It is a common occur- 
rence to find that two contractors may be working on the same 
kind of material and under precisely similar conditions and at 
the same price, and yet one may be making money and the other 
losing it — all on account of difference of management. 

116. Limit of profitable haul. As intimated in §§ 103 and 
110, there is with every method of haul a limit of distance be- 
yond which the expense for excessive hauling will exceed the 
loss resulting from borrowing and wasting. This distance is 
somewhat dependent on local conditions, thus requiring an inde- 
pendent solution for each particular case, but the general prin- 
ciples involved will be about as follows: Assume that it has been 
determined, as in Fig. 62, that the cut and fill wiR exactly bal- 
ance between two points, as between e and x, assuming that, as 
indicated in § 101 (9), a trestle has been introduced between s 
and t, thus altering the mass curve to Estxn . . . Since there 
is a balance between A' and C, the material for the fill between 
C and 6' must be obtained either by ^'borrowing'' in the im- 
mediate neighborhood or by transportation from the excavation 



138 RAILROAD CONSTRUCTION. § 116. 

between z' and n' , If cut and fill have been approximately 
balanced in the selection of grade line, as is ordinarily done, 
borrowing material for the fill CV im Dlies a wastage of material 
at the cut z'n' , To compare the two methods, we may place 
against the plan of borrowing and wasting, (a) cost, if an}'-, of 
extra right of way that may be needed from which to obtain 
earth for the fill C'e'\ (h) cost of loosening, loading, hauling 
a distance equal to that between the centers of gravity of the 
borrow-pit and of the fill, and the other expenses incidental to 
borro^\^ng M cubic yards for the fill Ce'; (c) cost of loosening, 
loading, hauling a distance equal to that between the centers 
of gravity of the cut z^n^ and of the spoil-bank, and the other 
expenses incidental to wasting M cubic yards at the cut z^n^; 
(d) cost, if any, of land needed for the spoil-bank. The cost of 
the other plan Avill be the cost of loosening, loading, hauling (the 
hauling being represented by the trapezoidal figure Cexn), and 
the other expenses incidental to making the fill Ce' with the 
material from the cut z^n^, the amount of material being M cubic 
3^ards, which is represented in the figure by the vertical ordi- 
nate from e to the line Cn. The difference between these costs 
will be the cost, if any, of land for borrow-pit and spoil-bank 
plus the cost of loosening, loading, etc. (except hauling and 
roadways) of M cubic yards, minus the difference in cost of the 
excessive haul from Ce to xn and the comparatively short hauls 
from borrow-pit and to spoil-bank. 

As an illustration, taking some of the estimates previously 
given for operating with average material, the cost of all items, 
except hauling and roadways, would be about as follows: 
loosening, with plough, 1.2 c, loading 5.0 c, spreading 1.5 c, 
wear, depreciation, etc., .25 c, superintendence, etc., 1.5 c; 
total 8.95 c. Suppose that the haul for both borrowing and 
wasting averages 100 feet or 1 station. Then the cost of haul 
per yard, using carts, would be (§ 109, a) [125X3(l + 4)]-^600 
= 3.125 c. The cost of roadways would be about 0.1 c. per 3^ard, 
making a total of 3.225 c. per cubic yard. Assume 31 = 10000 
cubic yards and the area (7e.rn = 180000 yards-stations or the 
equivalent of 10000 yards hauled 1800 feet. This haul would 
cost [125X3(18 + 4)]--600 = 13.75 c. per cubic yard. The cost 
of roadways will be 18 X .1 or 1.8 c, making a total of 15.55 c. for 
hauling and roadways. The difference of cost of hauling and 
roadways v/ill be 15.55 — (2X3.225) =9.10 c. per yard or $910 



§ 117. EARTHWORK. 139 

for the 10000 yards. Offsetting this is the cost of loosening, etc, 
10000 yards, at 8.95 e., costing $895. These figures may be 
better compared as follows: 



Long Haul. ^ 



f Loosening, etc., 10000 yards, (^ 8.95 c. $ 895. 

! Hauling, ** 10000 " @, 15.55 c. 1555. 



$2450. 



f Loosening, etc., 10000 yards (borrowed), @ 8.95 c. $895. 



10000 " (wasted), @ 8.95 c. 895. 

I Hauling, etc., 10000 " (borrowed), @ 3.225 c. 322.50 

i " " 10000 " (wasted), ^ 3.225 c. 322.50 

I $2435.00 



These costs are practically balanced, but no allowance has 
been made for right of way. If any considerable amount had 
to be paid for that, it would decide this particular case in favor 
of the long haul. This shows that under these conditions 1800 
feet is about the limit of profitable haul, the land costing nothing 
extra. 

BLASTING. 

117. Explosives. The effect of blasting is due to the ex- 
tremely rapid expansion of a gas which is developed by the 
decomposition of a very small amount of solid matter. Blasting 
compounds may be divided into two general classes, (a) slow- 
burning and (6) dQtonating. Gunpowder is a type of the slow- 
burning compounds. These are generally ignited by heat; the 
ignition proceeds from grain to grain; the heat and pressure 
produced are comparatively low. Nitro-glycerine is a type of 
the detonating compounds. They are exploded by a shock 
which instantaneously explodes the whole mass. The heat and 
pressure developed are far in excess of that produced by the 
explosion of powder. Nitro-glycerine is so easily exploded 
that it is very dangerous to handle. It was discovered that if 
the nitro-glycerine was absorbed by a spongy material like infu- 
sorial earth, it was much less liable to explode, while its poAver 
when actually exploded was practically equal to that of the 
amount of pure nitro-glycerine contained in the dynamite, whicli 
is the name given to the mixture of nitro-glycerine and infusorial 
earth. Nitro-glycerine is expensive; many other explosive 
chemical compounds which properly belong to the slow-burning 



140 RAILROAD CONSTRUCTION. § 118. 

class are comparatively cheap. It has been conclusively demon- 
strated that a mLxture of nitro-glycerine and some of the cheaper 
chemicals has a greater explosive force than the sum of the 
strengths of the component parts when exploded separately. 
Whatever the reason, the fact seems established. The reason is 
possibly that the explosion of the nitro-glycerine is sufhciently 
powerful to produce a detonation of the other chemicals, which 
is impossible to produce by ordinary means, and that this explo- 
sion caused by detonation is more poAverful than an ordinary 
explosion. The majority of the explosive compounds and 
^' powders'^ on the market are of this character — a mixture of 
20 to 60 per cent, of nitro-glycerine with variable proportions of 
one or more of a great variety of explosive chemicals. 

The choice of the explosive depends on the character of the 
rock. A hard brittle rock is most effectively blasted by a 
detonating compound. The rapidity with which the full force 
of the explosive is developed has a shattering effect on a brittle 
substance. On the contrary, some of the softer tougher rocks 
and indurated clays are but little affected by dynamite. The 
result is but little more than an enlargement of the blast-hole. 
QuarrAnng must generally be done with blasting-powder, as the 
quicker explosives are too shattering. Although the results 
obtained by various experimenters are very variable, it may be 
said that pure nitro-gl3xerine is eight times as powerful as black 
powder, dynamite (75% nitro-glycerine) six times, and gun- 
cotton four to six times as powerful. For open work where 
time is not particularl}^ valuable, black powder is by far the 
cheapest, but in tunnel-headings, whose progress determines the 
progress of the whole work, d^mamite is so much more effective 
and so expedites the work that its use becomes economical. 

1 1 8. Drilling. Although many very complicated forms of 
drill-bars have been devised, the best form (with slight modifi- 
cations to s\iit circumstances) is as shown in Fig. 64, (a) and (5). 
The width should flare at the bottom (a) about 15 to 30%. For 
hard rock the curve of the edge should be somewhat flatter and 
for soft rock somewhat more curved than shown. Fig. 64, (a). 
Sometimes the angle of the two faces is varied from that given, 
Fig. 64, (6), and occasionally the edge is purposely blunted so 
as to give a crushing rather than a cutting effect. The drills 
will require sharpening for each 6 to 18 inches depth of hole, 
and will require a new edge to be worked every 2 to 4 days. 



§ 119. 



EARTHWORK. 



141 



For drilling vertical holes the churn-drill is the most econom- 
ical. The drill-bar is of iron, about 6 to 8 feet long, 1^" in 
diameter, weighs about 25 to 30 lbs., and is shod with a piece 
of steel welded on. The bar is lifted a few inches between each 
blow, turned partially around, and allowed to fall, the impact 
doing the work. From 5 to 15 feet of holes, depending on the 
character of the rock, is a fair day's work — 10 hours. In very 
soft rocks even more than this may be done. This method is 





Fig. 64. 



inapplicable for inclined holes or even for vertical holes in con- 
fined places, such as tunnel-headings. For such places the only 
practical hand method is to use hammers. This may be done 
by light drills and light hammers (one-man work), or by heavier 
drills held by one man and struck by one or two men with heav}^ 
hammers. The conclusion of an exhaustive investigation as to 
the relative economy of light or heavy hammers is that the light- 
hammer method is more economical for the softer rocks, the 
heavy-hammer method is more economical for the harder rocks, 
but that the light-hammer method is always more expeditious 
and hence to be preferred when time is important. 

The subject of machine rock-drills is too vast to be treated 
here. The method is only practicable when the amount of 
work to be done is large, and especially when time is valuable. 
The machines are generally operated by compressed air for tun- 
nel-work, thus doing the additional service of supplying fresh 
air to the tunnel-headings where it is most needed. The cost 
per foot of hole drilled is quite variable, but is usually some- 
what less than that of hand-drilling — sometimes but a small 
fraction of it. 

119. Position and direction of drill-holes. As the cost of 
drilling holes is the largest single itc^in in the total cost of blast- 
ing, it is necessary that skill and judgment should be used in so 



142 



RAILROAD CONSTRUCTION. 



§ 120. 



locating the holes that the blasts will be most effective. The 
greatest effect of a blast will evidently be in the direction of the 
"line of least resistance." In a strictly homogeneous material 
this will be the shortest hne from the center of the explosive to 
the surface. The variations in homogeneity on account of 
laminations and seams require that each case shall be judged 
according to experience. In open-pit blasting it is generally 
easy to obtain two and sometimes three exposed faces to the 

rock, making it a simple matter 
to drill holes so that a blast will 
do effective work. When a soHd 
face of rock must be broken into, 
as in a tunnel-heading^ the work 
is necessarily ineffectual and ex- 
pensive. A conical or wedge- 
shaped mass will first be blown 
out by simultaneous blasts in 
the holes marked 1, Fig. 65; 
blasts in the holes marked 2 and 
3 will then complete the cross- 
section of the heading. A great saving in cost may often be 
secured by skilfully taking advantage of seams, breaks, and irreg- 
ularities. When the work is economically done there is but little 
noise or throwing of rock, a covering of old timbers and branches 
of trees generally sufficing to confine the smaller pieces which 
would otherwise fly up. 

120. Amount of explosive. The amount of explosive required 
varies as the cube of the line of least resistance. The best 
results are obtained when the line of least resistance is | of the 
depth of the hole ; also when the powder fills about J of the hole. 
For average rock the amount of powder required is as follows : 




DRILL HOLES IN T.UNNEL HEADING 
Fig. 65. 



Line of least resistance, 
Weight of powder 



2 ft. 


4 ft. 


6 ft. 


i lb. 


2 lbs. 


6i lbs. 



8 ft. 
16 Ib^ 



Strict compliance with all of the above conditions would re- 
quire that the diameter of the hole should vary for every case. 
W^hile this is impracticable, there should evidently be some 
variation in the size of the hole, depending on the work to be 
done. For example, a F' hole, drilled 2' 8'' deep, with its 
line of least resistance 2', and loaded with i lb, of powder, would 



§ 121. EARTHWORK. 143 

be filled to a depth of 9J'', which is nearly i of the depth. A 
3" hole, drilled 8' deep, with its line of least resistance 6', and 
loaded with 6J lbs. of powder, would be filled to a depth of over 
28'', which is also nearly J of the depth. One pound of blasting- 
powder will occupy about 28 cubic inches. Quarrying necessi- 
tates the use of numerous and sometimes repeated light charges of 
powder, as a heavy blast or a powerful explosive like dynamite 
is apt to shatter the rock. This requires more powder to the 
cubic yard than blasting for mere excavation, which may usually 
be done by the use of | to |- lb. of powder per cubic yard of easy 
open blasting. On account of the great resistance offered by 
rock when blasted in headings in tunnels, the powder used per 
cubic yard will run up to 2, 4, and even 6 lbs. per cubic yard. 
As before stated, nitro-glycerine is about eight times (and 
dynamite about six times) as powerful as the same weight of 
powder. 

121. Tamping. Blasting-powder and the slow-burning ex- 
plosives require thorough tamping. Clay is probably the best, 
but sand and fine powdered rock are also used. Wooden plugs, 
inverted expansive cones, etc., are periodically reinvented by 
enthusiastic inventors, only to be discarded for the simpler 
methods. Owing to the extreme rapidity of the development 
of the force of a nitro-glycerine or dynamite explosion, tamping 
is not so essential with these explosives, although it unquestion- 
ably adds to their effectiveness. Blasting under water has been 
effectively accomplished by merely pouring nitro-glycerine into 
the drilled holes through a tube and then exploding the charge 
without any tamping except that furnished by the superincum- 
bent Avater. It has been found that air-spaces about a charge 
make a material reduction in the effectiveness of the explosion. 
It is therefore necessary to carefully ram the explosive into a 
solid mass. Of course the liquid nitro-gh'cerine needs no ram- 
ming, but dynamite should be rammed with a wooden rammer. 
Iron should be carefully avoided in ramming gunpowder. A 
copper bar is generally used. 

122. Exploding the charge. Black powder is generally ex- 
ploded by means of a fuse which is essentially a cord in which 
there is a thin vein of gunpowder, the cord being protected by 
tar, extra linings of hemp, cotton, or even gutta-percha. The 
fuse is inserted into the middle of the charge, and the tampmg 
carefully packed around it so that it will not be injured. To 



144 RAILROAD CONSTRUCTION. § 123. 

produce the detonation required to explode nitro-glycerine and 
dynamite, there must be an initial explosion of some easily 
ignited explosive. This is generally accomplished by means of 
caps containing fulminating-powder which are exploded by 
electricity. The electricity (in one class of caps) heats a very 
fine platinum wire to redness, thereby igniting the sensitive 
powder, or (in another class) a spark is caused to jump through 
the powder between the ends of two wires suitably separated. 
Dynamite can also be exploded by using a small cartridge of 
gunpowder which is itself exploded by an ordinary fuse. 

123. Cost. Trautwine estimates the cost of blasting (for 
mere excavation) as averaging 45 cents per cubic yard, falling 
as low as 30 cents for easy but brittle rock, and running up to 
60 cents and even $1 when the cutting is shallow, the rock 
especially tough, and the strata unfavorably placed. Soft tough 
rock frequently requires more powder than harder brittle rock. 

124. Classification of excavated material. The classification 
of excavated material is a fruitful source of dispute between 
contractors and railroad companies, owing mainly to the fact 
that the variation between the softest earth and the hardest rock 
is so gradual that it is very difficult to describe distinctions 
between different classifications which are unmistakable and 
indisputable. The classification frequently used is (a) earth, 
(b) loose rock, and (c) sohd rock. As blasting is frequently 
used to loosen ''loose rock" and even ''earth" (if it is frozen), 
the fact that blasting is employed cannot be used as a criterion, 
especially as this would (if allowed) lead to unnecessary blasting 
for the sake of classifying material as rock. 

Earth. This includes clay, sand, gravel, loam, decomposed 
rock and slate, boulders or loose stones not greater than 1 cubic 
foot (3 cubic feet, P. R. R.), and sometimes even "hard-pan." 
In general it will signify material which can be loosened by a 
plough with two horses, or with w^hich one picker can keep one 
shoveller busy. 

Loose rock. This includes boulders and loose stones of more 
than one cubic foot and less than one cubic yard; stratified rock, 
not more than six inches thick, separated by a stratum of clay; 
also all material (not classified as earth) which may be loosened 
by pick or bar and which " can be quarried without blasting, 
although blasting may occasionally be resorted to," 



n^ 



EARTHAYORK. 145 



Solid rock includes all rock found in masses of over one cubic 
yard which cannot be removed except by blasting, 
B It is generally specified that the engineer of the railroad 
company shall be the judge of the classification of the material, 
but frequently an appeal is taken from his decisions to the 
courts. 

125. Specifications for earthwork. The following specifica- 
tions, issued by the Norfolk and lYestern R R., represent the 
average requirements. It should be remembered that very 
strict specifications invariably increase the cost of the work, 
and frequently add to the cost more than is gained by improved 
quality of work. 

1. The grading will be estimated and paid for by the cubic 
yard, and will include clearing and grubbing, and all open ex- 
cavations, 'channels, and embankments required for the forma- 
tion of the roadbed, and for turnouts and sidings; cutting all 
ditches or drains about or contiguous to the road; digging the 
foundation-pits of all culverts, bridges, or walls; reconstructing 
turnpikes or common roads in cases where they are destroyed or 
interfered with; changing the course or channel of streams; and 
all other excavations or embankments connected with or incident 
to the construction of said Railroad. 

2. All grading, except where otherwise specified, whether 
for cuts or fills, will be measured in the excavations and will be 
classified under the following heads, viz.: Solid Rock, Loose 
Rock, Hard-pan, and Earth. 

Solid Rock shall include all rock occurring in masses which, 
in the judgment of the said Engineer Maintenance of Way, may 
be best removed by blasting. 

Loose Rock shall include all kinds of shale, soapstone, and 
other rock which, in the judgment of the said Engineer Main- 
tenance of Way, can be removed by pick and bar, and is soft and 
loose enough to be removed without blasting, although blasting 
may be occasionally resorted to ; also, detached stone of less than 
one (1) cubic yard and more than one (1) cubic foot. 

Hard-pan shall consist of tough indurated clay or cemented 
gravel, which requires blasting or other equally expensive 
means for its remo^^al, or which cannot be ploughed with less 
than four horses and a railroad plough, or which requires two 
pickers to a shoveller, the said Engineer Maintenance of Way 
to be the judge of these conditions. 



146 RAILROAD CONSTRUCTION. § 125. 

Earth shall include all material of an earthy nature, of what- 
ever name or character, not unquestionably loose rock or hard- 
pan as above defined. 

Powder. The use of powder in cuts will not be considered 
as a reason for any other classification than earth, unless the 
material in the cut is clearly other than earth under the above 
specifications. 

3. Earth, gravel, and other materials taken from the exca- 
vations, except w^hen otherwise directed by the said Engineer 
Maintenance of Way or his assistant, shall be deposited in the 
adjacent embankment; the cost of removing and depositing 
which, when the distance necessary to be hauled is not more 
than sixteen hundred (1600) feet, shall be included in the price 
paid for the excavation. 

4. Extra Haul will be estimated and paid for as follows: 
whenever material from excavations is necessarily hauled a 
greater distance than sixteen hundred (1600) feet, there shall be 
paid in addition to the price of excavation the price of extra 
haul per 100 feet, or part thereof, after the first 1600 feet; the 
necessary haul to be determined in each case by the said Engi- 
neer Maintenance of Way or his assistant, from the profile and 
cross-sections, and the estimates to be in accordance therewith. 

5. All embankments shall be made in layers of such thick- 
ness and carried on in such manner as the said Engineer Mainte- 
nance of Way or his assistant may prescribe, the stone and heavy 
materials being placed in slopes and top. And in completing 
the fills to the proper grade such additional heights and fulness 
of slope shall be given them, to provide for their settlement, as 
the said Engineer Maintenance of Way, or his assistant, may 
direct. Embankments about masonry shall be built at such 
times and in such manner and of such materials as the said Engi- 
neer Maintenance of Way or his assistant may direct. 

6. In procuring materials for embankments from without 
the line of the road, and in wasting materials from cuttings, the 
place and manner of doing it shall in each case be indicated by 
the Engineer Maintenance of Way or his assistant; and care 
must be taken to injure or disfigure the land as little as possible. 
Borrow-pits and spoil-banks must be left by the Contractor in 
regular and sightly shape. 

7. The lands of the said Railroad Company shall be cleared 
to the extent required by the said Engineer Maintenance of 



I 



§125. EARTHWORK. 147 

Way, or his assistant, of all trees, brushes, logs, and other perish- 
able materials, which shall be destroyed by burning or deposited 
in heaps as the said Engineer Maintenance of Way, or his assist- 
ant, may direct. Large trees must be cut not more than two 
and one-half (2 J) feet from the ground, and under embank- 
ments less than four (4) feet high they shall be cut close to the 
ground. All small trees and bushes shall be cut close to the 
ground. 

8. Clearing shall be estimated and paid for by the acre or 
fraction of an acre. 

9. All stumps, roots, logs, and other obstructions shall be 
grubbed out, and removed from all places where embankments 
occur less than two (2) feet in height; also, from all places where 
excavations occur and from such other places as the said Engi- 
neer Maintenance of Way or his assistant may direct. 

10. Grubbing shall be estimated and paid for by the acre or 
fraction of an acre. 

11. Contractors, when directed by the said Engineer Main- 
tenance of Way or his assistant in charge of the work, will deposit 
on the side of the road, or at such convenient points as may be 
designated, any stone, rock, or other materials that they may 
excavate; and all materials excavated and deposited as above, 
together with all timber removed from the line of the road, will 
be considered the property of the Railroad Company, and the 
Contractors upon the respective sections will be responsible for 
its safe-keeping until removed by said Railroad Company, or 
until their work is finished. 

12. Contractors will be accountable for the maintenance of 
safe and convenient places wherever public or private roads are 
in any way interfered with by them during the progress of the 
work. They Avill also be responsible for fences thrown down, 
and for gates and bars left open, and for all damages occasioned 
thereby. 

13. Temporary bridges and trestles, erected to facilitate the 
progress of the work, in case of delays at masonry structures 
from any cause, or for other reasons, ^^'ill be at the expense of 
the Contractor. 

14. The line of road or the gradients may be changed in any 
manner, and at any time, if the said Engineer Maintenance of 
Way or kis assistant shall consider such a change necessary or 
expedient; but no claim for an increase in prices of excavation 



148 RAILROAD CONSTRUCTION. § 125. 

or embankment on the part of the Contractor will be allowed 
or considered unless made in writing before the work on that 
part of the section where the alteration has been made shall have 
been commenced. The said Engineer Maintenance of Way or 
his assistant may also, on the conditions last recited, increase or 
diminish the length of any section for the purpose of more nearly 
equalizing or balancing the excavations and embankments, or 
for any other reason. 

15. The roadbed will be graded as directed by the said En- 
gineer Maintenance of Way or his assistant, and in conformity 
with such breadths, depths, and slopes of cutting and filling as 
he may prescribe from time to time, and no part of the work 
will be finally accepted until it is properly completed and dressed 
off at the required grade. 



CHAPTER IV. 
TRESTLES. 

126. Extent of use. Trestles constitute from 1 to 3% of the 
length of the average railroad. It was estimated in 1889 that 
there was then about 2400 miles of single-track railway trestle 
in the United States, divided among 150,000 structures and esti- 
mated to cost about $75,000,000. The annual charge for main- 
tenance, estimated at J of the cost, therefore am.ounted to about 
$9,500,000 and necessitated the annual use of perhaps 300,000,000 
ft. B. M. of timber. The corresponding figures at the present 
time must be somewhat in excess of this. The magnitude of 
this use, which is causing the rapid disappearance of forests, has 
resulted in endeavors to limit the use of timber for this purpose. 
Trestles may be considered as justifiable under the following 
conditions : 

a. Permanent trestles. 

1. Those of extreme height — then called viaducts and fre- 
quently constructed of iron or steel, as the Kinzua viaduct, 302 
ft. high. 

2. Those across waterways — e.g.^ that across Lake Pontchar- 
train, near New Orleans, "22 miles lon^, 

3. Those across swamps of soft deep mud, or across a river- 
bottom, liable to occasional overflow. 

h. Temporary trestles. 

1. To open the road for traffic as quickly as possible — often 
a reason of great financial importance. 

2. To quickly replace a more elaborate structure, destroyed 
by accident, on a road already in operation, so that the inter- 
ruption to traffic shall be a minimum. 

3. To form an earth embankment with earth brought from 
a distant point by the train-load, when such a measure would 
cost less than to borrow earth in the immediate neighborhood. 

4. To bridge an opening temporarily and thus allow time to 
learn the regimen of a stream in order to better proportion the 

149 



150 RAILROAD CONSTRUCTION. § 127. 

size of the waterway and also to facilitate bringing suitable stone 
for masonry from a distance. In a new country there is always 
the double danger of either building a culvert too small, requir- 
ing expensive reconstruction, perhaps after a disastrous washout, 
or else wasting money b}^ constructing the culvert unnecessarily 
large. Much masonry has been built of a very poor quality of 
stone because it could be conveniently obtained and because 
good stone was unobtainable except at a prohibitive cost for 
transportation. Opening the road for traffic by the use of 
temporar}^ trestles obviates both of these difficulties. 

127. Trestles vs. embankments. Low embankments are very 
much cheaper than low trestles both in first cost and mainte- 
nance. Very high embankments are very expensive to con- 
struct, but cost comparatively little to maintain. A trestle of 
equal height ma}^ cost much less to construct, but will be expen- 
sive to maintain — perhaps' f of its cost per year. To determine 
the height beyond which it will be cheaper to maintain a trestle 
rather than build an embankment, it will be necessary to allow 
for the cost of maintenance. The height will also depend on 
the relative cost of timber, labor, and earthwork. At the pres- 
ent average values, it will be found that for less heights than 
25 feet the first cost of an embankment will generally be. less 
than that of a trestle; this implies that a permanent trestle 
should never be constructed with a height less than 25 feet except 
for the reasons given in § 126. The height at Avhich a permanent 
trestle is certainly cheaper than earthwork is more uncertain. 
A high grade line joining two hills will invariably imply at least 
a culvert if an embankment is used. If the culvert is built of 
masonry, the cost of the embankment will be so increased that 
the height at which a trestle becomes economical will be mate- 
rially reduced. The cost of an embankment increases much 
more rapidly than the height — with very high embankments 
more nearly as the square of the height — while the cost of 
trestles does not increase as rapidly as the height. Although 
local circumstances may modify the application of any set rules, 
it is probably seldom that it will be cheaper to build an embank- 
ment 40 or 50 feet high than to permanently maintain a wooden 
trestle of that height. A steel viaduct would probably be the 
best solution of such a case. These are frequently used for 
permanent structures, especially when ver}^ high. The cost of 
maintenance is much less than that of wood, which makes the 



§ 128. TRESTLES. 151 

use of iron or steel preferable for permanent trestles unless wood 
is abnormally cheap. Neither the cost nor the construction 
of iron or steel trestles will be considered in this chapter. 

128. Two principal types. There are two principal types of 
wooden trestles — pile trestles and framed trestles. The great 
objection to pile trestles is the rapid rotting of the portion of the 
pile which is underground, and the difficulty of renewal. The 
maximum height of pile trestles is about 30 feet, and even this 
height is seldom reached. Framed trestles have been con- 
structed to a height of considerabh^ over 100 feet They are 
frequently built in such a manner that any injured piece may be 
readily taken out and renewed without interfering with traffic. 
Trestles consist of two parts — the supports called ^^ bents/' and 
the stringers and floor system. As the stringers and floor system 
are the same for both pile and framed trestles, the ^^ bents" are 
all that need be considered separately. 

PILE TRESTLES. 

129. Pile bents. A pile bent consists generally of four piles 
driven into the ground deep enough to afford not only sufficient 
vertical resistance but also lateral resistance. On top of these 
piles is placed a horizontal ^^cap." The caps are fastened to 
the tops of the piles by methods illustrated in Fig. 66. The 
method of fastening sho^\Ti in each case should not be considered 
as applicable only to the particular t3^pe of pile bent used to illus- 
trate it. Fig. 66 (a and d) illustrates a mortise- joint with a hard" 
wood pin about 1^ in diameter. The hole for the pin should 
be bored separately through the cap and the mortise, and the 
hole through the cap should be at a slightly higher level than 
that through the mortise, so that the cap will be drawn down 
tight when the pin is driven. Occasionally an iron dowel (an 
iron pin about Ih" in diameter and about 6'' long) is inserted 
partly in the cap and partty in the pile. The use of drift-bolts, 
shown in Fig. 66 (6), is cheaper in first cost, but renders repairs 
and renewals very troublesome and expensive. ^' Split caps,*' 
shown in Fig. 66 (c), are formed by bolting two half -size strips 
on each side of a tenon on top of the pile. Repairs are very 
easily and cheaply made without interference with the traffic 
and without injuring other pieces of the bent. The smaller 
pieces are more easily obtainable in a sound condition; the 



152 



RAILROAD CONSTRUCTION. 



§129. 



decay of one does not affect the other, and the first cost is but 
little if any greater than the method of using a single piece. For 
further discussion, see § 136. 

For very hght traffic and for a height of about 5 feet three 
vertical piles will suffice; as shown in Fig. 66 (a)o Up to a height 




Fig. 66. 



of 8 or 10 feet four piles may be used without sway-bracing, as 
in Fig 66 (b), if the piles have a good bearing. For heights 
greater than 10 feet sway-bracing is generally necessary. The 
outside piles are frequently driven with a batter varying from 
1 : 12 to 1 ! 4. 

Piles are made, if possible, from timber obtained in the 
vicinity of the work. Durability is the great requisite rather 
than strength, for almost any timber is strong enough (except 
as noted below) and will be suitable if it will resist rapid decay. 
The following list is quoted as being in the order of preference 
on account of durability 



Red cedar 
Red cypress 
Pitch-pine 
Yellow pine 



White pine 
Redwood 
Elm 
Spruce 



9. White oak 

10. Post-oak 

11. Red oak 



12. Black oak 

13. Hemlock 

14. Tamarac 



Red-cedar piles are said to have an average life of 27 years 
with a possible maximum of 50 years, but the timber is rather 



§ 130. TRESTLES. 153 

weak, and if exposed in a river to flowing ice or driftwood is 
apt to be injured. Under these circumstances oak is prefer- 
able, although its life may be only 13 to 18 years. 

130. Methods of driving piles. The foUo^ving are the prin- 
cipal methods of driving piles : 

a. A hammer weighing 2000 to 3000 lbs. or more, sliding 
in guides, is drawn up by horse-power or a portable engine, and 
allowed to fall freely. 

h. The same as above except that the hammer does not fall 
freely, but drags the rope and revolving drum as it falls and is 
thus quite materially retarded. The mechanism is a little more 
simple, but is less effective, and is sometimes made deliberately 
deceptive by a contractor by retarding the blow, in order to 
apparently indicate the requisite resistance on the part of 
the pile. 

The above methods have the advantage that the mechanism 
is cheap and can be transported into a new country vdih com- 
parative ease, but the work done is somewhat ineffective and 
costly compared with some of the more elaborate methods 
given below. 

c. Gunpowder pile-drivers, which automatically explode a 
cartridge every time the hammer falls. The explosion not only 
forces the pile doT\TL, but throws up the hammer for the next 
blow. For a given height of fall the effect is therefore doubled. 
It has been shown by experience, however, that when it is at- 
tempted to use such a pile-driver rapidly the mechanism be- 
comes so heated that the cartridges explode prematurely, and the 
method has therefore been abandoned. 

d. Steam pile-drivers, in which the hammer is operated 
directly by steam. The hammer falls freely a height of about 
40 inches and is raised again by steam. The effectiveness is 
largely due to the rapidity of the blows, which does not allow 
time between the blows for the ground to settle around the pile 
and increase the resistance, which does happen when the blows 
are infrequent. ^'The liammer-cyHnder weighs 5500 lbs., and 
with 60 to 75 lbs. of steam gives 75 to 80 blows per minute. 
With 41 blows a large unpointed pile was driven 35 feet into a 
hard clay bottom in half a minute." Such a driver would cost 
about $800. 

The above four methods are those usual for dry earth. In 
very soft wet or sandy soils, where an unlimited supply of water 



154 RAILROAD CONSTRUCTION. § 131. 

is available, the water-jet is sometimes employed. A pipe is 
fastened along the side of the pile and extends to the pile-point. 
If water is forced through the pipe, it loosens the sand around 
the point and, rising along the sides, decreases the side resist- 
ance so that the pile sinks by its ovna weight, aided perhaps by 
extra weights loaded on. This loading may be accomplished by 
connecting the top of the pile and the pile-driver by a block 
and tackle so that a portion of the weight of the pile-driver is 
continually thrown on the pile. 

Excessive driving frequently fractures the pile below the 
surface and thereby greatly weakens its bearing power. To 
prevent excessive ^'brooming'' of the top of the 
pile, owing to the action of the hammer, the top 
should be protected by an iron ring fitted to the 
top of the pile. The ^'brooming'' not only ren- 
ders the driving ineffective and hence uneconomi- 
cal, but vitiates the value of any test of the bearing 
power of the pile by noting the sinking due to a 
given weight falling a given distance. If the pile 
is so soft that brooming is unavoidable, the top 
Fig. 67. should be adzed off frequently, and especially 
should it be done just before the final blows which are to test its 
bearing-power. 

In a new country judgment and experience mil be required 
to decide intelligently whether to employ a simple drop-hammer 
machine, operated by horse-power and easily transported but 
uneconomical in operation, or a more complicated machine 
working cheaply and effectively after being transported at 
greater expense. 

131. Pile- driving formulae. If R = ihe resistance of a pile, 
and s the set of the pile during the last blow, w the weight of 
the pile-hammer, and h the fall during the last blow, then we 

may state the approximate relation that Rd==wh, or R = — . 

This is the basic principle of all rational formulae, but the maxi- 
mum weight which a pile will sustain after it has been driven 
some time is by no means equal to the resistance of the pile 
during the last blow. There are also many other modifying 
elements which have been A^ariously allowed for in the many 
proposed formula). The formula range from the extreme of 
empirical simplicity to very complicated attempts to allow 




§131. TRESTLES. 155 

properly for all modifying causes. As the simplest rule, speci- 
fications sometimes require that the piles shall be driven until 
the pile will not sink more than 5 inches under five consecutive 
blows of a 2000-lb. hammer falling 25 feet. The ^^Engineering 

News formula" * gives the safe load as r, in which iv = 

s + 1 

weight of hammer, h=iall in feet, s=set of pile in inches under 
the last blow. This formula is derived from the above basic 
formula by calling the safe load ^ of the final resistance, and 
by adding (arbitrarily) 1 to the final set (s) as a compensation 
for the extra resistance caused by the settling of earth around 
the pile between each blow. This formula is used only for 
ordinary hammer-driving. When the piles are driven by a 

steam pile-driver the formula becomes safe load = ^— r. «For 

^ s + 0.1 

the '' gunpowder pile-driver,'' since the explosion of the cartridge 

drives the pile in with the same force with which it throws the 

hammer upward, the effect is twice that of the fall of the hammer, 

^wh 
and the formula becomes safe load = - — ---. In these last two 

s + 0.1 

formulae the constant in the denominator is changed from s + 1 

to s + 0.1. The constant (1.0 or 0.1) is supposed to allow, as 

before stated, for the effect of the extra resistance caused by the 

earth settling around the pile between each blow. The more 

rapid the blows the less the opportunity to settle and the less 

the proper value of the constant. 

The above formulae have been given on account of their 
simplicity and their practical agreement with experience. Many 
other formulae have been proposed, the majority of which are 
more complicated and attempt to take into account the weight of 
the pile, resistance of the guides, etc. While these elements, 
as well as many others, have their influence, their effect is so 
overshadowed by the indeterminable effect of other elements — 
as, for example, the effect of the settlement of earth around the 
pile between blows — that it is useless to attempt to employ any- 
thing but a purely empirical formula. 

Examples. 1. A pile was driven with an ordinary hammer 
weighing 2500 pounds until the sinking under five consecutive 
blows was 15i inches. The fall of the hammer during the last 



* Engineering News, Nov. 17, 1892. 



156 



RAILROAD CONSTRUCTION. 



§ 132. 



blows was 24 feet. What was the safe bearing power of the 
pile? 



2wh 2X2500X24 120000 
s + l'"(iXl5.5) + l"" 4.1 



= 29300 pounds. 



2. Piles are being driven into a firm soil with a steam pile- 
driver until they show a safe bearing power of 20 tons. The 
hammer weighs 5500 pounds and its fall is 40 inches. What 
should be the sinking under the final blow? 



40000 



2wh 2X5 500 X 3^ 
s + 0.1~ s + 0.1 ' 



36667 



40000 



0.1 =.81 inch. 



\*%/V, 



132. Pile-points and pile-shoes. Piles are generally sharpened 
to a blunt point. If the pile is liable to strike boulders, sunken 

logs, or other obstructions which are 
liable to turn the point, it h necessary 
to protect the point by some form of 
shoe. Several forms in cast iron have 
been used, also a wrought-iron shoe, 
having four ^'straps'' radiating from 
the apex, the straps being nailed on to 
the pile, as shown in Fig. 68 (6). The 
cast-iron form shown in Fig. 68 (a) 
has a base cast around a drift-bolt. 
The recess on the top of the base re- 
PiQ- 68. ceiA'CS the bottom of the pile and pre- 

vents a tendency to split the bottom of the pile or to force the 
shoe off laterally. 

133. Details of design. No theoretical calculations of the 
strength of pile bents need be attempted on account of the ex- 
treme complication of the theoretical strains, the uncertainty as 
to the real strength of the timber used, the variability of that 
strength with time, and the insignificance of the economy that 
would be possible even if exact sizes could be computed. The 
piles are generally required to be not less than 10'' or 12" in 
diameter at the large end. The P. R. R. requ res that they shall 




§ 134. TRESTLES 157 

be " not less than 14 and 7 inches in diameter at butt and small 
end respectively, exclusive of bark, which must be removed/' 
The removal of the bark is generally required in good Avork. 
Soft durable woods, such as are mentioned in § 129, are best 
for the piles, but the caps are generally made of oak or yellow 
pine. The caps are generally 14 feet long (far single track) 
with a cross-section 12''XlV' or i2''Xl4". ^^ Split caps" 
would consist of two pieces 6"Xl2''. The sway-braces, never 
used for less heights than G', are made of 3''Xl2'' timber, and 
are spiked on with f spikes W long. The floor system w^ill be 
the same as that described later for framed trestles. 

134. Cost of pile trestles. The cost, per linear foot, of piling 
depends on the method of driving, the scarcity of suitable tim- 
ber, the price of labor, the length of the piles, and the amount 
of siiifting of the pile-driver required. The cost of soft-wood 
piles varies from 8 to 15 c. per lineal foot, and the cost of oak 
piles A^aries from 10 to 30 c. per foot according to the length, 
the longer piles costing more per foot. The cost of driving Avill 
average about S2.o0 per pile, or 7.5 to 10 c. per lineal foot. 
Since the cost of shifting the pile-driver is quite an item in the 
total cost, the cost of driving a long pile would be less per foot 
than for a short pile, but on the other hand the cost of the pile 
is greater per foot, which tends to make the total cost per foot 
constant. Specifications generally say that the piling will be 
paid for per lineal foot of piling left in the work. The wastage 
of the tops of piles salved off is always something, and is fre- 
quently very large. Sometimes a small amount per foot of 
piling sawed off is allowed the contractor as compensation for 
his loss. This reduces the contractor's risk and possibly reduces 
his bid by an equal or greater amount than the extra amount 
actually paid him. 



FRAMED TRESTLES. 

135. Typical design. A typical design for a framed trestle 
bent is given in Fig. 69. This represents, with slight variations 
of detail, the plan according to AA^hich a large part of the framed 
trestle bents of the country have been built — i.e., of those less 
than 20 or 30 feet in height, not requiring multiple story con- 
struction. 

136. Joints, (a) The mortise-and- tenon joint is illustrated in 



158 



RAILROAD CONSTRUCTION. 



§ 136. 



Fig. 69 and also in Fig. 66 (a). The tenon should be about 




FiCx. 69. 



AA/\^ 



\\^ 




Fig. 70. 



3'' thick, 8" wdde, and 5^'' long. The mortise should be cut 
a little deeper than the tenon. ^'Drip-holes'' 
from the mortise to the outside will assist in 
draining off water that ma.y accumulate in the 
joint and thus prevent the rapid decay that 
would otherwise ensue. These joints are very 
troublesome if a single post deca3's and requires 
renewal. It is generally required that the mor- 
tise and tenon should be thoroughly daubed 
with paint before putting them together. This will tend to 
make the joint water-tight and prevent decay from the accu- 
mulation and retention of water in the joint. 

(b) The plaster joint. This joint is made by bolting and 
spiking a 3''Xl2'' plank on 
both sides of the joint. The 
cap and sill should be 
notched to receive the posts. 
Repairs are greatly facili- 
tated by the use of these 
joints. This method has been 
used by the Delaware and 
Hudson Canal Co. [R. R.]. 

(c) Iron plates. An iron plate of the form shown in Fig. 72 




Fig. 71. 



f 



§137. 



TRESTLES. 



159 



Fig. 72 (a). Bolts passing 




/'" 


o o 


b 

c 


o 
o 




o 
o 


a 


o o 


(b) " 

a 



Fig. 72. 



apply with even greater force to 



(b) is bent and used as shown in 
through the bolt - holes 
shown secure the plates j 

to the timbers and make 
a strong joint which may 
be readily loosened for re- 
pairs. By slight modifi- 
'cations in the design the 
method may be used for 
inclined posts and compli- 
cated joints. 

(d) Split caps and sills. 
These are described in 
§ 129. Their advantages 
framed trestles. 

(e) Dowels and drift-bolts. These joints facilitate cheap and 
rapid construction, but renewals and repairs are very difficult, it 
being almost impossible to extract a drift-bolt, vrhich has been 
driven its full length, without splitting open the pieces contain- 
ing it. Notwithstanding this objection they are extensively 
used, especially for temporary work which is not expected to 
be used long enough to need repairs. 

137. Multiple-story construc- 
tion. Single-story framed trestle 
bents are used for heights up 
to 18 or 20 feet and exception- 
ally up to 30 feet. For greater 
heights some such construction as 
is illustrated in a skeleton design 
in Fig. 73 is used. By using split 
sills between each story and sepa- 
rate vertical and batter posts in 
each story, any piece may readil}' 
be removed and renewed if neces- 
sary. The height of these stories 
varies, in different designs, from 
15 to 25 and even 30 feet. In 
some designs the structure of each 
story is independent of the stories 
above and below. This greatly 
facilitates both the original construction and subsequent repairs. 




160 



RAILROAD CONSTRUCTION. 



§138. 



In other designs the verticals and batter-posts are made con- 
tinuous through two consecutive stories. The structure is 
somewhat stiffer, but is much more difficult to repair. 

Since the bents of any trestle are usually of variable height 
and those heights are not always an even multiple of the uniform 
height desired for the stories, it becomes necessary to make the 




Fig. 74. 

upper stories of uniform height and let the odd amount go to the 
lowest story, as sho^Ti in Figs. 73 and 74. 

138. Span. The shorter the span the greater the number of 
trestle bents; the longer the span the greater the required strength 
of the stringers supporting the floor. Economy demands the 
adoption of a span that shall make the sum of these require- 



OtJ 




Fig. 75, 



ments a minimum. The higher the trestle the greater the cost 
of each bent, and the greater the span that would be justifiable. 
Nearly all trestles have bents of variable height, but the advan- 
tage of employing uniform standard sizes is so great that many 



§ 139, 



TRESTLES. 



161 




Wi'4 



roads use the same span and sizes of timber not only for the 
panels of any given trestle, but also for all trestles regardless of 
height. The spans generally used vary from 10 to 16 feet. The 
Norfolk and Western R. R. uses a span of 12' 6" for all single- 
story trestles, and a span of 25' for all multiple-story trestles. 
The stringers are the same in both cases, but when the span is 
25 feet, knee-braces are run from the sill of the first story below 
to near the middle of each set of stringers. These knee-braces 
are connected at the top by a "straining-beam" on which the 
stringers rest, thus supporting the stringer in the center and vir- 
tually reducing the span about one-half. 

139. Foundations, (a) Piles. Piles are frequently used as a 
foundation, as in Fig. 76, particularly in soft ground, and also 
for temporar}^ structures. These 
foundations are cheap, quickly 
constructed, and are particularly 
valuable when it is financially 
necessary to open the road for 
traffic as soon as possible and 
with the least expenditure of 
money; but there is the disad- 
vantage of inevitable decay 
within a few years unless the piles are chemicallj^ treated, as will 
be discussed later. Chemical treatment, however, increases the 
cost so that such a foundation would often cost more than a 
foundation of stone. A pile should be driven under each post 
as shown in Fig. 76. 

(b) Mud-sills. Fig. 77 illustrates the use of mud-sills as 

built by the Louisville and 
Nashville R. R. Eight blocks 
12"X12"X6' are used under 
each bent. When the ground 
is very soft, two additional 
timbers (12" X 12" X length of 
bent-sill), as shown by the 
dotted lines, are placed under- 
neath. The number required 
evidently depends on the na- 

'^^^•^^* ture of the ground. 

(c) Stone foundations. Stone foundations are the best and 
the most expensive. For very high trestles the Norfolk and 



EiG. 76. 



n \\n \\ 



\--\ 1-- 


"^-iE]--i: 



162 



RAILROAD CONSTRUCTION. 



§ 140. I 



Western R. R. employs foundations as shown in Fig. 78, the 
walls being 4 feet thick. When the height of the trestle is 72 
feet or less (the plans requiring for 72' in height a foundation- 
wall 39' 6" long) the foundation is made continuous. The sill 



SILL OF TRESTLE 




^^ 




< 8 *- ■< ^3 ^ 

Fig. 78. 



of the trestle should rest on several short lengths of 3"X12" 
plank laid transverse to the sill on top of the wall. 

140. Longitudinal bracing* This is required to give the 
structure longitudinal stiffness and also to reduce the columnar 
length of the posts. This bracing generally consists of hori- 
zontal ^' waling-strips'' and diagonal braces. Sometimes the 
braces are placed wholly on the outside posts unless the trestle 
is very high. For single-story trestles the P. R. R. employs 
the ^' laced'' system, i.e., a line of posts joining the cap of one 
bent with the sill of the next, and the sill of that bent with the 
cap of the next. Some plans employ braces forming an X in 
alternate panels. Connecting these braces in the center more 
than doubles their columnar strength. Diagonal braces, when 
bolted to posts, should be fastened to them as near the ends of 
the posts as possible. The sizes employed vary largely, depend- 
ing on the clear length and on whether they are expected to act 
by tension or compression. 3''Xl2" planks are often used 
when the design would require tensile strength only, and 8" X 8" 
posts are often used when compression may be expected. 

141. Lateral bracing. Several of the more recent designs of 
trestles employ diagonal lateral bracing between the caps of 
adjacent bents. It adds greatly to the stiffness of the trestle 
and better maintains its alignment. 6"X6" posts, forming 
an X and connected at the center, will answer the purpose. 

142. Abutments. When suitable stone for masonry is at 
hand and a suitable subsoil for a foundation is obtainable without 
too much excavation, a masonry abutment will be the best. 
Such an abutment would probably be used when masonry foot- 
ings for trestle bents were employed (§ 139, c). 

Another method is to construct a "crib" of 10"Xl2" timber^ 



§ 143. 



TRESTLES. 



163 



laid horizontally, drift-bolted together, securely braced and 
embedded into the ground. Except for temporary construction 
such a method is generally 
objectionable on account of 
rapid decay. 

Another method, used most 
commonly for pile trestles, and 
for framed trestles having pile 
foundations (§ 139, a), is to use 
a pile bent at such a place that 
the .natural surface on the up- 
hill side is not far below the 
cap, and the thrust of the material, filled in to bring the surface 
to grade, is insignificant. 3'''Xl2'' planks are placed behind 
the piles, cap, and stringers to retain the filled material. 




Fig. 79. 



FLOOR SYSTEMS. 

143. Stringers. The general practice is to use two, three, 
and even four stringers under each rail. Sometimes a stringer 
is placed under each guard-rail. Generally the stringers are 
made of two panel lengths and laid so that the joints alternate. 
A few roads use stringers of only one panel length, but this prac- 
tice is strongly condemned by many engineers. The stringers 
should be separated to allow a circulation of air around them 
and prevent the decay which would occur if they were placed 
close together. This is sometimes done by means of 2'' planks, 
6' to 8' long, which are placed over each trestle bent. Several 
bolts, passing through all the stringers forming a group and 
through the separators, bind them all into one solid construc- 
tion. Cast-iron ^^ spools" or washers, varying from 4'' to f" 
in length (or thickness), are sometimes strung on each bolt so 
as to separate the stringers. Sometimes washers are used 
between the separating planks and the stringers, the object of 
the separating planks then being to bind the stringers, especially 
abutting stringers, and increase their stiffness. 

The most common size for stringers is S'^XlG''. The Penn- 
sylvania Railroad varies the width, depth, and number of 
stringers under each rail according to the clear span. It may 
be noticed that, assuming a uniform load per running foot, both 
the pressure per square inch at the ends of the stringers (the 



164 



EAILROAD CONSTRUCTION. 



§144. 



caps having a width of 12") and also the stress due to trans- 
verse strain are kept approximately constant for the variable 
gross load on these varying spans. 



Clear span. 


No. of pieces 
under each rail. 


Width. 


Depth. 


10 feet 
12 " 
14 '• 
16 •• 


2 
2 
2 
3 


8 inches 
8 *' 
10 " 
8 •• 


15 inches 

16 " 

17 •* 
17 *• 



144. Corbels. A corbel (in trestle-work) is a stick of timber 
(perhaps two placed side by side), about 3' to 6' long, placed 
underneath and along the stringers and resting on the cap. 
There are strong prejudices for and^ against their use, and a 
corresponding diversity in practice. They are bolted to the 
stringers and thus stiffen the joint. They certainly reduce the 
objectionable crushing of the fibers at each end of the stringer, 
but if the corbel is no wider than the stringers, as is generally 
the case, the area of pressure between the corbels and the cap is 




Fig. 80. 

no greater and the pressure per square inch on the cap is no less 
than the pressure on the cap if no corbels were used. If the 
corbels and cap are made of hard wood, as is recommended by 
some, the danger of crushing is lessened, but the extra cost and 
the frequent scarcity of hard wood, and also the extra cost and 
labor of using corbels, may often neutralize the advantages 
obtained by their use. 

145. Guard-rails. These are frequently made of 5''X8'' stuff, 
notched V for each tie. The sizes vary up to 8"X8", and the 
depth of notch from f to IJ''. They are generally bolted to 
every third or fourth tie. It is frequently specified that they 
shall be made of oak, white pine, or yellow pine. The joints- 
are made over a tie, by halving each piece, as illustrated in Fig. 
81. The joints on opposite sides of the trestle should be ^'stag- 



§ 146. TRESTLES. 165 

gered.^' Some roads fasten every tie to the guard-rail, using a 
bolt, a spike, or a lag-screw. 

Guard-rails were originally used ^dth the idea of preventing 
the wheels of a derailed truck from running off the ends of the 
ties. But it has been found that an outer guard-rail alone (with- 
out an inner guard-rail) becomes an actual element of danger, 
since it has frequently happened that a derailed wheel has caught 
on the outer guard-rail, thus causing the truck to slew around 




Fig. 81. 

and so produce a dangerous accident. The true function of the 

outside guard-rail is thus changed to that of a tie-spacer, which 
keeps the ties from spreading when a derailment occurs. The 
inside guard-rail generally consists of an ordinary steel rail 
spiked about 10 inches inside of the running rail. These inner 
guard-rails should be bent inward to a point in the center of the 
track about 50 feet beyond the end of the bridge or trestle. If 
the inner guard-rails are placed with a clear space of 10 inches 
inside the running rail, the outer guard-rails should be at least 
6' 10'' apart. They are generally much farther apart than this. 

146. Ties on trestles. If a car is derailed on a bridge or 
trestle, the heavily loaded wheels are apt to force their way be- 
tween the ties by displacing them unless the ties are closely 
spaced and fastened. The clear space between ties is generally 
equal to or less than their width. Occasionally it is a little more 
than their width. 6''X8'' ties, spaced 14" to 16'' from center 
to center, are most frequently used. The length varies from 
9' to 12' for single track. They are generally notched J" deep 
on the under side where they rest on the stringers. Oak ties 
are generally required even when cheaper ties are used on the 
other sections of the road. Usually every third or fourth tic is 
bolted to the stringers. When stringers are placed underneath 
the guard-rails, bolts are run from the top of the guard-rail to 
the under side of the stringer. The guard-rails thus hold down 
the whole system of ties, and no direct fastening of the ties to 
the stringers is needed. 

147. Superelevation of the outer rail on curves. The location 
of curves on trestles should be avoided if possible, especially 
when the trestle is high. Serious additional strains are intro- 



166 



RAILROAD CONSTRUCTION. 



147. 



duced especially when the curvature is sharp or the speed high. 
Since such curves are sometimes practically unavoidable, it is 
necessary to design the trestle accordingly. If a train is stopped 
on a curved trestle, the action of the train on the trestle is 
evidently vertical. If the train is moving with a considerable 
velocity, the resultant of the weight and the centrifugal action 
is a force somewhat inclined from the vertical. Both of these 
conditions may be expected to exist at times. If the axis of 
the system of posts is vertical (as illustrated in methods a, &, c, c?, 
and e), any lateral force, such as would be produced by a mov- 
ing train, Avill tend to rack the trestle bent. If the stringers are 
set vertically, a centrifugal force likewise tends to tip them 
side wise. If the axis of the S3^stem of posts (or of the stringers) 
is inclined so as to coincide with the pressure of the train on the 
trestle when the train is moving at its normal velocity, there is 
no tendency to rack the trestle when the train is moving at that 
velocity, but there will be a tendency to rack the trestle or 
twist the stringers when the train is stationar}^ Since a moving 
train is usually the normal condition of affairs, as well as the 
condition w^hich produces the maximum stress, an inclined axis 
is evidently preferable from a theoretical standpoint ; but what- 
ever design is adopted, the trestle should evidently be suffi- 
ciently cross-braced for either a moving or a stationary load, 
and any proposed design must be studied as to the effect of both 
of these conditions. Some of the various methods of securing 
the requisite superelevation may be described as follows : 

(a) Framing the outer posts longer than the inner posts, so 

that the cap is inclined at the 
proper angle; axis of posts verti- 
cal. (Fig. 82.) The method re- 
quires more work in framing the 
trestle, but simphfies subsequent 
track-laying and maintenance, un- 
less it should be found that the 
superelevation adopted is unsuit- 
able, in which case it could be cor- 
rected by one of the other methods 
given below. The stringers tend 
to twist when the train is sta- 
tionaiy. 

(b) Notching the cap so that the stringers are at a different 




Fig. 82. 



147. 



TRESTLES. 



167 



Fig. 83. 



elevation. (Fig. 83.) This weakens the cap and requires that 
all ties shall be notched to a 
bevelled surface to fit the string- 
ers, which also weakens the ties. 
A centrifugal force will tend to 
twist the stringers and rack the 
trestle. 

(c) Placing wedges underneath 
the ties at each stringer. These 
wedges are fastened Avith two 
bolts. Two or more wedges will 
be required for each tie. The ad- 
ditional number of pieces required 
for a long curve will be immense, and the work of inspection and 
keeping the nuts tight will greatly increase the cost of main- 
tenance. 

(d) Placing a wedge under the outer rail at each tie. This 
requires but one extra piece per tie. There is no need of a 
wedge under the inn^r tie in order to make the rail normal to 
the tread. The resulting inward inclination is substantially that 
produced by some forms of rail-chairs or tie-plates. The spikes 
(a little longer than usual) are driven through the wedge into 
the tie. Sometimes '^lag-screws" are used instead of spikes. 
If experience proves that the superelevation is too much or too 
little, it may be changed by this method with less work than 
by any other. 

(e) Corbels of different heights. When corbels are used (see 

§ 144) the required in- 
clination of the floor sys_ 
tem may be obtained by 
varying the depth of the 
corbels. 

(f) Tipping the whole 
trestle. This is done by 
placing the trestle on an 
inclined foundation. If 
very much inclined, the 
trestle bent must be se- 
cured against the possi- 
bility of slipping sidewise, 
for the slope would be considerable with a sharp curve, and the 




Fig. 84. 



168 RAILROAD CONSTRUCTION. § 148. 

vibration of a moving train would reduce the coefficient of 
friction to a comparatively small quantity. 

(g) Framing the outer posts longer. This case is identical 
with case (a) except that the axis of the system of posts is 
inclined, as in case (/), but the sill is horizontal. 

The above-described plans will suggest a great variety of 
methods which are possible and which differ from the above 
only in minor details. 

148. Protection from fire. Trestles are peculiarly subject to 
fire, from passing locomotives, which may not only destroy the 
trestle, but perhaps cause a terrible disaster. This danger is 
sometimes reduced by placing a strip of galvanized iron along 
the top of each set of stringers and. also along the tops of the 
caps. Still greater protection was given on a long trestle on the 
Louisville and Nashville R. R. by making a solid flooring of 
timber, covered with a layer of ballast on which the ties and 
rails were laid as usual. 

Barrels of water should be provided and kept near all trestles^ 
and on very long trestles barrels of water should be placed every 
two or three hundred feet along its length. A place for the bar- 
rels may be provided by using a few ties which have an extra 
length of about four feet, thus forming a small platform, which 
should be surrounded by a railing. The track-walker should be 
held accountable for the maintenance of a supply of water in 
these barrels, renewals being frequently necessary on account of 
evaporation. Such platforms should also be provided as refuge- 
bays for track- walkers and trackmen working on the trestle. On 
very long trestles such a platform is sometimes provided with 
sufficient capacity for a hand-car. 

149. Timber. Any strong durable timber may be used when 
the choice is limited, but oak, pine, or cypress are preferred 
when obtainable. When all of these are readily obtainable, 
the various parts of the trestle will be constructed of different 
kinds of wood — the stringers of long-leaf pine, the posts and 
braces of pine or red cypress, and the caps, sills, and corbels (if 
used) of white oak. The use of oak (or a similar hard wood) 
for caps, sills, and corbels is desirable because of its greater 
strength in resisting crushing across the grain, which is the | 
critical test for these parts. There is no ph3^siological basis to 
the objection, sometimes made, that different species of timber, 
in contact with each other, will rot quicker than if only one 




t Id 

25'o"CENTtR TO CENT| •? j 

FOR TWO STORY b'enT rri I 

12'6"CENTERT0 GEN"* | j 
FOR ONE STORV BENTt 



NORFOLK & WESTER | 

STANDARD ONE AND TWO STQ j 
i (SEPT. 10,1S91.) i 

\ I ^ U- 

n 




i I 



t PLUMB POSTS, SINGLE E 

i " " IN LOWER STORY. DOUBLE 

PERS, SINGLE 

" " " DOUBLE 

rTER POST, EXCEPT IN 8"x 1 a'lNTERMEDlAT 

TO LENGTHS GIVEN BY ABOVE FORMULAS. 




§ 150. TRESTLES. 169 

kind of timber is used. AVlien a very extensive trestle is to be 
built at a place where suitable growing timber is at hand but 
there is no convenient sawmill, it will pay to transport a port- 
able sawmill and engine and cut up the timber as desired. 

150. Cost of framed timber trestles. The cost varies widely 
on account of the great variation in the cost of timber. When 
a railroad is first penetrating a new and undeveloped region, the 
cost of timber is frequently small, and when it is obtainable from 
the company's right-of-way the only expense is felling and 
sawing. The work per M, B. M., is small, considering that a 
single stick 12'' X 12'' X 25' contains 300 feet, B. M., and that 
sometimes a few hours' work, worth less than $1, will finish all 
the work required on it. Smaller pieces will of course require 
more work per foot, B. M. Long-leaf pine can be purchased 
from the mills at from $8 to $12 per M feet, B. M., according 
to the dimensions. To this must be added the freight and labor 
of erection. The cartage from the nearest railroad to the trestle 
may often be a considerable item. Wrought iron will cost 
about 3 c. per pound and cast iron 2 c, although the prices are 
often lower than these. The amount of iron used depends on 
the detailed design, but, as an average, will amount to $1.50 
to $2 per 1000 feet, B. M., of timber. A large part of the tres- 
tling of the country has been built at a contract price of about 
$30 per lOQO feet, B. M., erected. While the cost will frequently 
rise to $40 and even $50 when timber is scarce, it will drop to 
$13 (cost quoted) when timber is cheap. 

DESIGN OF WOODEN TRESTLES. 

151. Common practice. A great deal of trestling has been 
constructed without any rational design except that custom and 
experience have shown that certain sizes and designs are probably 
safe. This method has resulted occasionally in failures but more 
frequently in a very large waste of timber. Many railroads 
employ a uniform size for all posts, caps, and sills, and a uniform 
size for stringers, aU regardless of the height or span of the 
trestle. For repair work there are practi(%l reasons favoring 
this. "To attempt to run a large lot of sizes would be more 
wasteful in the end than to maintain a few stock sizes only. 
Lumber can be bought more cheaply by giving a general order 
for ' the run of the mill for the season,' or * a cargo lot/ specify- 



170 RAILROAD CONSTRUCTION. § 152. 

ing approximate percentages of standard stringer size, of 
12 X 12-inch stuff, 10 X 10-inch stuff, etc., and a Hberal propor- 
tion of 3- or 4-inch plank, all lengths thrown in. The 12 X 12- 
inch stuff, etc., is ordered all lengths, from a certain specified 
length up. In case of a wreck, washout, burn-out, or sudden 
call for a trestle to be completed in a stated time, it is much 
more economical and practical to order a certain number of 
carloads of 'trestle stuff' to the ground and there to select piece 
after piece as fast as needed, dependent only upon the length of 
stick required. When there is time to make the necessary sur- 
veys of the ground and calculations of strength, and to wait for a 
special bill of timber to be cut and delivered, the use of differ- 
ent sizes for posts in a structure would be warranted to a certain 
extent." * For new construction, when there is generally 
sufficient time to design and order the proper sizes, such waste- 
fulness is less excusable, and under any conditions it is both 
safer and more economical to prepare standard designs which 
can be made applicable to varying conditions and which will at 
the same time utilize as much of the strength of the timber as 
can be depended on. In the following sections will be given 
the elements of the preparation of such standard designs, which 
will utilize uniform sizes with as little waste of timber as possible. 
It is not to be understood that special designs should be made 
for each individual trestle. 

152. Required elements of strength. The stringers of trestles 
are subject to transverse strains, to crushing across the grain 
at the ends, and to shearing along the neutral axis. The strength 
of the timber must therefore be computed for all these kinds 
of stress. Caps and sills will fail, if at all, by crushing across 
the grain; although subject to other forms of stress, these could 
hardly cause failure in the sizes usually employed. There is an 
apparent exception to this: if piles are improperly driven and 
an uneven settlement subsequently occurs, it may have the 
effect of transferring practically all of the weight to two or three 
piles, while the cap is subjected to a severe transverse strain 
which may cause its failure. Since such action is caused gener- 
ally by avoidable erijprs of construction it may be considered as 
abnormal, and since such a failure will generally occur by a 
gradual settlement, all danger may be avoided by reasonable 

* From "Economical Designing of Timber Trestle Bridges." 



§153. TRESTLES. 171 

care in inspection. Posts- must be tested for their columnar 
strength. These parts form the bulk of the trestle and are the 
parts which can be definitely designed from known stresses. 
The stresses in the bracing are more indefinite, depending on 
indeterminate forces, since the inclined posts take up an un- 
known proportion of the lateral stresses, and the design of the 
bracing may be left to what experience has shown to be safe, 
without invohdng any large waste of timber. 

153. Strength of timber. Until recently tests of the strength 
of timber have generally been made by testing small, selected, 
well-seasoned sticks of ^^ clear stuff," free from knots or imper- 
fections. Such tests would give results so much higher than 
the vaguely known strength of large unseasoned ^^ commercial" 
timber that very large factors of safety were recommended — • 
factors so large as to detract from any confidence in the whole 
theoretical design. Recently the U. S. Government has been 
making a thoroughly scientific test of the strength of full-size 
timber under various conditions as to seasoning, etc. The work 
has been so extensive and thorough as to render possible the 
economical designing of timber structures. 

One important result of the investigation is the determina- 
tion of the great influence of the moisture in the timber and 
the law of its effect on the strength. It has been also shown 
that timber soaked vdth. water has substantially the same 
strength as green timber, even though the timber had once been 
thoroughl}^ seasoned. Since trestles are exposed to the weather 
they should be designed on the basis of using green timber. 
It has been showm that the strength of green timber is very 
regularly about 55 to 60% of the strength of timber in which 
the moisture is 12% of the dry weight, 12% being the proportion 
of moisture usually found in timber that is protected from the 
weather but not heated, as, e.g., the timber in a barn. Since 
the moduli of rupture have aU been reduced to this standard of 
moisture (12%), if we take one-eighth of the rupture values, it 
still allows a factor of safety of about five, even on green timber. 
On page 172 there are quoted the values taken from the U. S. 
Government reports on the strength of timber, the tests probably 
being the most thorough and reliable that were ever made. 

On page 173 are given the " average safe allowable working 
unit stresses in pounds per square inch," as recommended by 
the committee on ^'Strength of Bridge and Trestle Timbers," 



172 



RAILROAD CONSTRUCTION. 



§ 154 j 



the work being done under the auspices of the Association of 
Railway Superintendents of Bridges and Buildings. The report 
was presented at their fifth annual convention, held in New 
Orleans, in October, 1895. 



Moduli of rupture for various timbers. [12% moisture.] 
(Condensed from U. S. Forestry Circular, No. 1.5.) 



No. 

1 

2 
3 
4 
5 
6 
7 



8 

9 

10 

II 
12 
13 
14 
15 
16 
19 
20 

21 
27 

28 
29 
30 



Species. 



Long-leaf 

Cuban 

Short-leaf 

Loblolly 

White 

Red 

Spruce 



pine 



Bald cypress . 
White cedar. 
Douglas snruce 



White oak 
Overcup " 
Post 
Cow 
Red 

Texan " 
Willow " 
Spanish " 



Shagbark hickory 
Pignut 

White elm 

Cedar " 

White ash 



u o 
« o 

M 



38 
39 
32 
33 
24 
31 
39 



29 
23 

32 



50 
46 
50 
46 
45 
46 
45 
46 



51 
56 
34 
46 
39 



Cross-bending. 






12 600 

13 600 

10 100 

11 300 
7 900 
9 100 

10 000 



7 900 

6 300 

7 900 

13 100 

11 300 

12 300 
11 500 

11 400 

13 100 
10 400 

12 000 



16 000 
18 700 
10 300 
13 500 
10 800 



Modulus 

of 
Elasticity. 



2 070 000 
2 370 000 

1 680 000 

2 050 000 
1 390 000 
1 620 000 
1 640 000 



1 290 000 

910 000 

1 680 000 



2 090 000 

1 620 000 

2 030 000 
1 610 000 
1 970 000 
1 860 000 
1 750 000 

1 930 000 

2 390 000 
2 730 000 
1 540 000 
1 700 000 
1 640 000 



Crush- 
ing 
end- 
wise. 



8000 
8700 
6500 
7400 
5400 
6700 
7300 



6000 
5200 
5700 i 



c3.S 

.S w) 

u 
O 



1180 
1220 

960 
1150 

700 
1000 
1200 



800 
700 
800 



c . 

'5 ^ 



700 
700 
700 
700 
400 
500 
800 



500 
400 
500 



8500 
7300 
7100 
7400 
7200 
8100 
7200 
7700 



9500 
10900 
6500 
8000 
7200 



2200 


1000 


1900 


1000 


3000 


1100 


1900 


900 


2300 


1100 


2000 


900 


1600 


900 


1800 


900 


2700 


1100 


3200 


1200 


1200 


800 


2100 


1300 


1900 


1100 



154. Loading. As sho^Ti in § 138, the span of trestles is always 
small, is generally 14 feet, and is never greater than 18 feet 
except when supported by knee-braces. The greatest load that 
will ever come on any one span will be the concentrated loading 
of the drivers of a consolidation locomotive. With spans of 14 
feet or less it is impossible for even the four pairs of drivers to 
be on the same span at once. The weight of the rails, ties, and 
guard-rails should be added to ol^tain the total load on the string- 
ers, and the weight of these, plus the weight of the stringers, 
should be added to obtain the pressure on the caps or corbels. 



§154, 



TRESTLES. 



173 






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174 



RAILROAD CONSTRUCTION. 



§155. 



This dead load is almost insignificant compared wdth the live 
load and may be included with it. The weight of rails, ties, 
etc., may be estimated at 200 pounds per foot. To obtain the 
weight on the caps the weight of the stringers must be added, 
which depends on the design and on the weight per cubic foot 
of the wood employed. But as the weight of the stringers is 
comparatively small, a considerable percentage of variation in 
weight will have but an insignificant effect on the result. Dis- 
regarding all refinements as to actual dimensions, the ordinary 
maximum loading for standard-gauge railroads may be taken 
as that due to four pairs of driving-axles, spaced 5' 0'' apart 
and giving a pressure of 25000 pounds per axle. This should 
be increased to 400C0 pounds per axle (same spacing) for the 
heaviest traffic. On the basis of 25000 pounds per axle the 
following results have been computed : 



STRESSES ON VARIOUS SPANS DUE TO MOVING LOADS OF 25000 

POUNDS, SPACED 5' 0'' APART. 


Span in feet. 


Max. moment, 
ft. lbs. 


Max. shear. 


Max. load on 
one cap. 


10 
12 
14 
16 
18 


65 000 
103 600 
142 400 
181 400 
220 600 


38 500 
45 000 
49 600 
54 725 
60 100 


52 100 
62 700 
74 200 
85 700 
97 900 



Although the dead load does not var}^ in proportion to the 
live load, yet, considering the very small influence of the dead 
load, there will be no appreciable error in assuming the corre- 
sponding values, for a load of 40000 lbs. per axle, to be |f of^ 
those given in the above tabulation. 

155. Factors of safety. The most valuable result of the gov-j 
ernment tests is the knowledge that under given moisture condi- 
tions the strength of A^arious species of sound timber is not the 
variable uncertain quantity it was once supposed to be, but that 
its strength can be relied on to a comparatively close percentage., 
This confidence in values permits the employment of lower fac- 
tors: of safety than have heretofore been permissible. Stresses, 
which when excessive would result in immediate destruction, 
such as cross-breaking and columnar stresses, should be allowed 
a higher factor of safety — say 6 or 8 for green timber. Other 
stresses, such as crushing across the grain and shearing along the 



§ 156. TRESTLES. 175 

neutral axis, which will be apparent to inspection before it is 
dangerous, may be allowed loAver factors — say 3 to 5. 

156. Design of stringers. The strength of rectangular beams 
of equal width varies as the square of the depth; therefore deep 
beams are the strongest. On the other hand, when any cross- 
sectional dimension of timber much exceeds 12'' the cost is 
much higher per M, B. M., and it is correspondingly difficult to 
obtain thoroughly sound sticks, free from wind-shakes, etc. 
Wind-shakes especially affect the shearing strength. Also, if 
the required transverse strength is obtained by using high nar- 
row stringers, the area of pressure between the stringers and the 
cap may become so small as to induce crushing across the grain. 
This is a very common defect in trestle design. As already in- 
dicated in § 138, the span should vary roughly with the average 
height of the trestle, the longer spans being employed when the 
trestle bents are very high, although it is usual to employ the 
same span throughout any one trestle. 

To illustrate, if we select a span of 14 feet, the load on one 
cap will be 74200 lbs. If the stringers and cap are made of 
long-leaf yellow pine, which require the closely determined value 
of 1180 lbs. per square inch to produce a crushing amounting 
to 3% of the height on timber with 12% moisture, we may use 
200 lbs. per square inch as a safe pressure even for green tim- 
ber; this will require 371 square inches of surface. If the cap 
is 12'' wide, this ^^dll require a width of 31 inches, or say 2 
stringers under each rail, each 8 inches wide. For rectangular 
beams 

Moment =iR'hh\ 

Using for R' the safe ^^alue 1275 lbs. per square inch, we have 

142400X 12 =iX 1275X32 X/i^ 
from which /i = 15".9. If desired, the width may be increased 
to 9" and the depth correspondingly reduced, which will give 
similarly /i = 14".9, or say 15". This shows that two beams, 
9"Xl5", under each rail will stand the transverse bending and 
have more than enough area for crushing. 
The shear per square inch wiU equal 

3 total shear 3 49600 , ^^ ,, 

— -. — = 77 -r:77r^7T^ = 138 lbs. per sq. mch, 

2 cross-section 24X9Xlo ^ ^ ' 

which is a safe value, although it should preferably be less 
Hence the above combination of dimensions will answer. 



176 HAILROAD CONSTRUCTION. § 157. 

The deflection should be computed to see if it exceeds the 
somewhat arbitrar}^ standard of g J^ of the span. The deflec- 
tion for uniform loading is 



5TFZ^ 

32WE ' 



in which Z= length in inches; 

TF = total load, assumed as uniform; ^ 

£' = modulus of elasticity, given as 2,070,000 lbs. 

per sq. in. for long-leaf pine, 12% dry, and assumed to be 
1,200,000 for green timber. Then 



5X^2800X 168^ 

32X36X15=^X1200000 



^^X168''=0''.84, 

so that the calculated deflection is well within the limit. Of 
course the loading is not strictly uniform, but even with a lib- 
eral allowance the deflection is still safe. 

For the heaviest practice (40000 lbs. per axle) these stringer 
dimensions must be correspondingly increased. 

157. Design of posts. Four posts are generally used for 
single-track work. The inner posts are usually braced by the 
cross-braces, so that their columnar strength is largely increased ; 
but as they are apt to get more than their share of work, the ad- 
vantage is compensated and they should be treated as unsup- 
ported columns for the total distance between cap and sill in 
simple bents, or for the height of stories in multiple-story con- 
struction. The caps and sills are assumed to have a width of 12". 
It facilitates the application of bracing to have the columns of 
the same width and vary the other dimension as required. 

Unfortunately the experimental work of the U. S. Govern- 
ment on timber testing has not yet progressed far enough to 
establish unquestionably a general relation between the strength 
of long columns and the crushing strength of short blocks. The 



§157. TRESTLES. 177 

following formula has been suggested, but it cannot be consid- 
ered as established: 

, ^^ 700 + 15c . , . , 

/=^><700 + 15^T7- '''''^^''^- 



/= allowable working stress per sq. in for long columns; 
F= " " '' '' '' '' '' short blocks; 

I 

Z= length of column in inches; 

cZ= least cross-sectional dimensions in inches. 

Enough work has been done to give great reliability to the two 
following formuljE for white pine and yellow pine, quoted from 
Johnson's "Materials of Construction," p. 684: 

1 /Z\ 2 
Working load per sq, in. =p = 1000 — ^ ( TT ) ' long-leaf pine ; 

'' " " ^^ =p= 600- i^'j-V, white pine; 

in which Z= length of column in inches, ai 

/i=least cross-sectional dimension in inches. 



The frequent practice is to use 12'' X 12'' posts for all trestles. 
If we substitute in the above formula Z =20' =240" and /i = 12" 
we have p = 1000 -K-¥/)' =900 lbs. 

900X144 = 129600 lbs., the working load for each post. This 
is more than the total load on one trestle bent and illustrates 
the usual great waste of timber. Making the post 8" X 12" and 
calculating similarly, we have p = 775, and the working load per 
column is 775X96 = 74400 lbs. As considerable must be 
allowed for "weathering," which destroys the strength of the 
outer layers of the wood, and also for the dynamic effect of 
the live load, 8" X 12" may not be too great, but it is certainly 
a safe dimension. 12" X 6" would possibly prove amply safe 
in practice. One method of allowing for weathering is to dis- 
regard the outer half -inch on all sides of the post, i.e., to cal- 
culate the strength of a post one inch smaller in each dimension 
than the post actually employed. On this basis an 8" X 12" X20' 



178 RAILROAD CONSTRUCTION. § 158. 

post, computed as a VxH' post, would have a safe columnar 
strength of 706 lbs. per square inch. With an area of 77 square 
inches, this gives a working load of 54362 lbs. for each post, or 
217448 lbs. for the four posts. Considering that 74200 lbs. 
is the maximum load on one cap (14 feet span), the great excess 
of strength is apparent. 

158. Design of caps and sills. The stresses in caps and sills 
are very indefinite, except as to crushing across the grain. As 
the stringers are placed almost directly over the inner posts, and 
as the sills are supported just under the posts, the transverse 
stresses are almost insignificant. In the above case four posts 
have an area of 4 X 12'' X 8'' =384 sq. in. The total load, 
74200 lbs.; will then give a pressure of 193 pounds per square 
inch, which is wdthin the allowable limit. This one feature 
might require the use of 8" X 12'' posts rather than 6" X 12" 
posts, for the smaller posts, although probably strong enough as 
posts, would produce an objectionably high pressure. 

159. Bracing. Although some idea of the stresses in the 
bracing could be found from certain assumptions as to wind- 
pressure, etc., 3'et it would probabl}^ not be found wdse to de- 
crease, for the sake of economy, the dimensions which practice 
has shown to be sufficient for the work. The economy that 
would be possible would be too insignificant to justify any risk. 
Therefore the usual dimensions, given in §§ 139 and 140, should 
be employed. 



CHAPTER V. 

TUNNELS. 
SURVEYING. 

1 60. Surface surveys. As tunnels are always dug from each 
end and frequently from one or more intermediate shafts, it is 
necessary that an accurate surface survey should be made 
between the two ends. As the natural surface in a locality 
where a tunnel is necessary is almost invariably very steep and 
rough, it requires the employment of unusually refined methods 
of work to avoid inaccuracies. It is usual to run a line on the 
surface that will be at every point vertically over the center line 
of the tunnel. Tunnels are generally made straight unless 
curves are absolutely necessary, as curves add greatly to the 
cost. Fig. 85 represents roughly a longitudinal section of the 




Fig. 



--€000 ^ -tOOO -r 600G r—- 50C^— i 

85. — Sketch of Section of the Hoosac Tunistel. 



Hoosac Tunnel. Permanent stations were located at A, B, (7, 
D, E, and F, and stone houses were built at A, B, C, and D. 
These were located with ordinary field transits at first, and then 
all the points were placed as nearly as possible in one vertical 
plane by repeated trials and minute corrections, using a very 
large specially constructed transit. The stations D and F were 
necessary because E and A were invisible from C and B. The 
alignment at A and E having been determined with great accu 
racy, the true alignment was easily carried into the tunnel. 

179 



180 RAILROAD CONSTRUCTION. § 161. 

The relative elevations of A and E were determined with 
great accuracy. Steep slopes render necessary many settings 
of the level per unit of horizontal distance and require that the 
work be unusually accurate to obtain even fair accuracy per 
unit of distance. The levels are usually re -run many times 
until the probable error is a very small quantity 

The exact horizontal distance between the two ends of the 
tunnel must also be known, especially if the tunnel is on a 
grade. The usual steep slopes and rough topography likewise 
lender accurate horizontal measurements very difficult. Fre- 
QLiently when the slope is steep the measurement is best ob- 
tained by measuring along the slope and allowing for grade. 
This may be very accurately done by employing two tripods 
(level or transit tripods serve the purpose very well), setting 
them up slightly less than one tape-length apart and measuring 
between horizontal needles set in wooden blocks inserted in the 
top of each tripod. The elevation of each needle is also observed. 
The true horizontal distance between two successive positions 
of the needles then equals the square root of the difference of 
the squares of the inclined distance and the difference of eleva- 
tion. Such measurements will probably be more accurate than 
those made by attempting to hold the tape horizontal and 
plumbing down with plumb-bobs, because (1) it is practically 
difficult to hold both ends of the tape trul}^ horizontal; (2) on 
steep slopes it is impossible to hold the down-hill end of a 100- 
foot tape (or even a 25-foot length) on a level with the other 
end, and the great increase in the number of applications of the 
unit of measurement very greatly increases the probable error 
of the whole measurement; (3) the vibrations of a plumb-bob 
introduce a large probability of error in transferring the meas- 
urement from the elevated end of the tape to the ground, and 
the increased number of such applications of the unit of meas- 
urement still further increases the probable error. 

i6i. Surveying down a shaft. If a shaft is sunk, as at S, 
Fig 85, and it is desired to dig out the tunnel in both directions 
from the foot of the shaft so as to meet the headings from the 
outside, it is necessary to know, when at the bottom of the 
shaft, the elevation, alignment, and horizontal distance from 
each end of the tunnel. • 

The elevation is generally carried down a shaft by means of 
a steel tape. This method involves the least number of appli- 



§ 161. TUNNELS. 181 

cations of the unit of measurement and greatly increases the 
accuracy of the final result. 

The horizontal distance from each end may be easily trans- 
ferred do^vn the shaft by means of a plumb -bob, using some of 
the precautions described in the next paragraph. 

To transfer the alignrnent from the surface to the bottom of 
a shaft requires the highest skill because the shaft is always 
small, and to produce a line perhaps several thousand feet long 
in a direction giA'en by two points 6 or 8 feet apart requires 
that the two points must be determined with extreme accuracy. 
The eminently successful method adopted in the Hoosac Tunnel 
will be briefly described: Two beams were securel}^ fastened 
across the top of the shaft (1030 feet deep), the beams being 
placed transversely to the direction of the tunnel and as far 
apart as possible and j^et allow plumb-lines, hung from the 
intersection of each beam with the tunnel center line, to swing 
freely at the bottom of the shaft. These intersections of the 
beams with the center line were determined by averaging the 
results of a large number of careful observations for alignment. 
Two fine parallel wires, spaced about ^\'^ apart, were then 
stretched between the beams so that the center line of fhe 
tunnel bisected at all points the space between the wires. 
Plumb-bobs, weighing 15 pounds, were suspended by fine wires 
beside each cross-beam, the wires passing between the two 
parallel alignment wires and bisecting the space. The plumb- 
bobs were allowed to swing in pails of water at the bottom. 
Drafts of air up the shaft required the construction of boxes 
surrounding the wires. Even these precautions did not suffice 
to absolutely prevent vibration of the wire at the bottom 
through a ^^ery small arc. The mean point of these vibrations 
in each case was then located on a rigid cross-beam suitably 
placed at the bottom of the shaft and at about the level of the 
roof of the tunnel. Short plumb-lines were then suspended 
from these points whenever desired; a transit was set (by trial) 
so that its line of collimation passed through both plumb-lines 
and the line at the bottom could thus be prolonged. 

Some recent experience in the ''Tamarack" shaft, 4250 feet 
deep, shows that the accuracy of the results may be affected by 
air-currents to an unsuspected extent. Two 50-lb. cast-iron 
plumb-bobs were suspended with No. 24 piano-wire in this 
shaft. The carefully measured distances between the wires 



182 



RAILROAD CONSTRUCTION. 



§162. 



at top and bottom were 16.32 and 16.43 feet respectively. 
After considerable experimenting to determine the cause of 
the variation, it was finally concluded that air-currents were 
alone responsible. The variation of the bobs from a true ver- 
tical plane passing through the wires at the top was of course 
an unknown quantity, but since the variation in one direction 
amounted to 0.11 foot, the accuracy in other directions was 
very questionable. This shows that a careful comparative 
measurement between the wires at top and bottom should 
always be made as a test of their parallelism. 

162. Underground surveys. Survey marks are frequently 
placed on the timbering, but they are apt to prove unreliable 
on account of the shifting of the timbering due to settlement 
of the surrounding material They should never be placed at 
the bottom of the tunnel on account of the danger of being 
disturbed or cohered up. Frequently holes are drilled in the 
roof and filled with wooden plugs in which a hook is screwed 
exactly on line Although this is probably the safest method, 
even these plugs are not always undisturbed, as the material, 
unless very hard, will often settle slightly as the excavation 
proceeds. When a tunnel is perfectly straight and not too long, 
alignment-points may be gixxn as frequently as desired from 

permanent stations located outside 
the tunnel where they are not liable 
to disturbance. This has been ac- 
complished by running the align- 
ment through the upper part of the 
cross-section, at one side of the cen- 
ter, where it is out of the way of 
the piles of masonry material, 
debris, etc., which are so apt to 
choke up the lower part of the 
cross-section. The position of this 
line relative to the cross-section 
being fixed, the alignment of any 
required point of the cross-section 
is readily found by means of a light 
frame or template with a fixed tar- 
get located where this line would intersect the frame when 
properly placed. A level-bubble on the frame will assist in 
setting the f rr.me in its proper position. 




I'iG. S6. 



§ 163. TUNNELS. 183 

In all tunnel surveying the cross-wires must be illuminated 
by a lantern, and the object sighted at must also be illuminated. 
A powerful dark-lantern with the opening covered with ground 
glass has been found useful. This may be used to illuminate a 
plumb-bob string or a very fine rod, or to place behind a brass 
plate having a narrow slit in it, the axis of the slit and plate 
being coincident with the plumb-bob string by which it is 
hung. 

On account of the interference to the surveying caused by 
the work of construction and also by the smoke and dust in the 
air resulting from the blasting, it is generally necessary to make 
the surve3^s at times when construction is temporarily sus- 
pended. 

163. Accuracy of tunnel surveying. Apart from the very 
natural desire to do surveying which shall check well, there is 
an important financial side to accurate tunnel surveying. If 
the survey lines do not meet as desired when the headings come 
together, it may be found necessary, if the error is of appreciable 
size, to introduce a slight curve, perhaps even a reversed curve, 
into the alignment, and it is even conceivable that the tunnel 
section would need to be enlarged som.ewhat to alloAv for these 
curves. The cost of these changes and the perpetual annoyance 
due to an enforced and undesirable alteration of the original 
design will justify a considerable increase in the expenses of the 
survey. Considering that the cost of surveys is usually but a 
small fraction of the total cost of the work, an increase of 10 or 
even 20% in the cost of the surveys will mean an insignificant 
addition to the total cost and frequently, if not generally, it will 
result in a saving of many times the increased cost. The 
accuracy actually attained in two noted American tunnels is 
given as follows: The Musconetcong tunnel is about 5000 feet 
long, bored through a mountain 400 feet high. The error of 
alignment at the meeting of the headings was 0'.04, error of 
levels 0'.015, error of distance 0'.52. The Hoosac tunnel is 
over 25000 feet long. The heading from the east end met the 
heading from the central shaft at a point 11274 feet from the 
east end and 1563 feet from the shaft. The error in alignment 
was -^\ of an inch, that of levels "a. few hundredths," error of 
distance ^'trifling." The alignment, corrected at the shaft, 
w^as carried on through and met the heading from the west end 
at a point 10138 feet from the west end and 2056 feet from 



184 RAILROAD CONSTRUCTION. § 164. I 

the shaft. Here the error of alignment was -/g'' and that of 
levels 0.134 ft. 

DESIGN. 

164. Cross-sections. Nearly all tunnels have cross-sections 
peculiar to themselves — all varying at least in the details. The 
general form of a great many tunnels is that of a rectangle sur- 
mounted by a semi-circle or semi-ellipse. In very soft material 
an inverted arch is necessary along the bottom. In such cases 
the sides Vvill generally be arched instead of vertical. The sides 
are frequently battered. With very long tunnels, several forms 
of cross-section will often be used in the same tunnel, owing to 
differences in the material encountered. In solid rock, which 
vdW. not disintegrate upon exposure, no lining is required, and 
the cross-section will be the irregular section left by the blasting, 
the only requirement being that no rock shall be left within the 
required cross-sectional figure. Farther on, in the same tunnel, 
when passing through some very soft treacherous material, it 
may be necessary to put in a full arch lining — top, sides, and 
bottom — which will be nearly circular in cross-section. For 
an illustration of this see Figs. 87 and 88. 

The ^^'idth of tunnels A^aries as greatly as the designs. Single- 
track tunnels generally have a Avidth of 15 to 16 feet. Occa- 
sionally they have been built 14 feet wide, and even less, and 
also up to 18 feet, especially when on curves. 24 to 26 feet is 
the most common width for double track. Many double-track 
tunnels are only 22 feet wide, and some are 28 feet wide. The 
heights are generally 19 feet for single track and 20 to 22 feet 
for double track. The variations from these figures are con- 
siderable. The lower limits depend on the cross -section of the 
rolling stock, with an indefinite allowance for clearance and ven- 
tilation. Cross-sections which coincide too closely with what is 
absolutely required for clearance are objectionable, because any 
slight settlement of the lining which would otherwise be harm- 
less would then become troublesome and even dangerous. Figs. 
87, 88, and 89 * shoAV some typical cross-sections. 

165. Grade. A grade of at least 2% is needed for drainage. 
If the tunnel is at the summit of two grades, the timnel grade 
should be practically level, Avith an alloAvance for drainage, the 

* Drinker's "Tunneling." 



§ 165. 



TUNNELS. 



186 




Fig. 87. — Hoosac Tunnel. Section through Solid Rock, 




Fig. 88. — Hoosac Tunnel. Section through Soft Ground. 



186 



RAILROAD CONSTRUCTION. 



§166. 



actual summit being perhaps in the center so as to drain both 
ways. When the tunnel forms part of a long ascending grade, 
it is advisable to reduce the grade through the tunnel unless the 
tunnel is a ery short The additional atmospheric resistance and 
the decreased adhesion of the dri^'er wheels on the damp rails in 
a tunnel will cause an engine to work very hard and still more 
rapidly vitiate the atmosphere until the accumulation of poison- 
ous gases becomes a source of actual danger to the engineer and 




St. Cloud Tunnel. 



fireman of the locomotive and of extreme discomfort to the 
passengers. If the nominal ruling grade of the road were 
maintained through a tunnel, the maximum resistance would be 
found in the tunnel. This would probabl}^ cause trains to stall 
there, which would be objectionable and perhaps dangerous. 

1 66. Lining. It is a characteristic of many kinds of rock 
and of all earthy material that, although they may be self- 
sustaining when first exposed to the atmosphere, they rapidly 
disintegrate and require that the top and perhaps the ?ides and 
even the bottom shall be lined to prevent caving in. In this 
country, when timber is cheap, it is occasionally framed as an 
arch and used as the permanent lining, but masonry is always 
to be preferred. Frequently the cross-section is made extra 



§167. 



TUNNELS, 



187 



large so that a masonry lining may subsequently be placed inside 
the wooden lining and thus postpone a large expense until the 
road is better able to pay for the work. In very soft unstable 
material, like quicksand, an arch of cut stone voussoirs may be 
necessary to withstand the pressure. A good quality of brick is 
occasionally used for lining, as they are easily handled and make 
good masonry if the pressure is not excessive. Only the best 
of cement mortar should be used, economy in this feature being 
the worst of folly. Of course the excavation must include the 
outside hue of the hning. Any excavation which is made out- 
side of this line (by the fall of earth or loose rock or by exces- 
sive blasting) must be refilled with stone well packed in. Occa- 
sionally it is necessary to fill these spaces with concrete. Of 
course it is not necessary that the lining be uniform throughout 
the tunnel. 

167. Shafts. Shafts are variously made with square, rectan- 
gular, eUiptical, and circular cross-sections. The rectangular 




Fig. 90. — Connection with Shaft, Church Hill Tunnel. 



cross-section, with the longer axis parallel with, the tunnel, is 
most usually employed. Generally the shaft is directly over the 
center of the tunnel, but that always implies a complicated con- 
nection between the linings of the tunnel and shaft, provided 



188 



RAILROAD CONSTRUCTION. 



§ 168. 



such linings are necessary. It is easier to sink a shaft near to 
one side of the tunnel and make an opening through the nearly 
vertical side of the tunnel. Such a method was employed in the 
Church Hill Tunnel, illustrated in Fig. 90.* Fig. 91 j shows 
a cross-section for a large main shaft. Many shafts have been 
built with the idea of being left open permanently for ventila- 
tion and have therefore been elaborately lined with masonry. 




Fig. 91. — Cross-section. Large Main Shaft. 

The general consensus of opinion now appears to be that shafts 
are worse than useless for ventilation; that the quick passage of 
a train through the tunnel is the most effective ventilator; and 
that shafts only tend to produce cross-currents and are ineffective 
to clear the air. In consequence, many of these elaborately 
lined shafts have been permanently closed, and the more recent 
practice is to close up a shaft as soon as the tunnel is completed. 
Shafts always form drainage -wells for the material they pass 
through, and sometimes to such an extent that it is a serious 
matter to dispose of the water that collects at the bottom, 
requiring the construction of large and expensive drains. 

1 68. Drains. A tunnel will almost invariably strike veins of 
water which will promptly begin to drain into the tunnel and 
not only cause considerable trouble and expense during construc- 
tion, but necessitate the provision of permanent drains for its 
perpetual disposal. These drains must frequently be so large as 

* Drinker's ''Tunneling." 

t Rziha, "Lehrbuch der Gesammten Tunnelbaukunst." 



m 



§ 169. 



TUNNELS. 



189 



to appreciably increase the required cross-section of the tunnel. 
Generally a small open gutter on each side will suffice for this 
purpose, but in double-track tunnels a large covered drain is 
often built between the tracks. It is sometimes necessary to 
thoroughly grout the outside of the lining so that water will not 
force its way through the masonry and perhaps injure it, but 
may freely drain doAATi the sides and pass through openings in 
the side walls near their base into the gutters. 



CONSTRUCTION. 

169. Headings. The methods of all tunnel excavation de- 
pend on the general principle that all earthy material, except 
the softest of liquid mud and quicksand, will be self-sustaining 
o^'er a greater or less area and for a greater or less time after 
excavation is made, and the work consists in excavating some 
material and immediately propping up the exposed surface by 
timbering and poling-boards. The excavation of the cross- 
section begins with cutting out a '^heading," which is a small 
horizontal drift whose breast is constanth^ kept 15 feet or more 
in adA'ance of the full cross-sectional excavation. In solid 
self-sustaining rock, which will not decompose upon exposure 
to air, it becomes simph^ a matter of excavating the rock with 
the least possible expenditure of time and energy. In soft 
ground the heading must be heavily timbered, and as the heading 
is gradually enlarged the timbering must be gradually extended 
and perhaps replaced, according to some regular system, so that 
when the full cross-section has been ex- 
cavated it is supported by such timbering 
as is intended for it. The heading is 
sometimes made on the center line near 
the top; with other plans, on the center 
line near the bottom; and sometimes two 
simultaneous headings are run in the two 
lo^^'er corners. Headings near the bot- 
tom serve the purpose of draining the 
material above it and facilitating the 
excavation. The simplest case of head- 
ing timbering is that shown in Fig. 92, 
in which cross-timbers are placed at in- p^^ g2. 

tervals just under the roof, set in notches 
cut in the side walls and supporting poling-boards which sus- 




190 



RAILROAD CONSTRTJCTIOK. 



§170. 



tain whatever pressure may come on them. Cross-timbers 
near the bottom support a flooring on ^^'hieh vehicles for trans- 
porting material may be run and under which the drainage 
may freely escape. As the necessity for timbering becomes 
greater, side timbers and even bottom timbers must be added, 
these timbers supporting poling-boards, and even the breast 
of the heading must be protected by boards suitably braced. 




Fig. 93. — Timbering for Tunnel Heading. 



as shown in Fig. 93. The supporting timbers are framed into 
collars in such a manner that added pressure only increases 
their rigidity. 

170. Enlargement. Enlargement is accomplished by remov- 
ing the pohng-boards, one at a time, excavating a greater or less 
amount of material, and immediately supporting the exposed 
material with poling-boards suitably braced. (See Figs. 93 and 
94.) This work being systematically done, space is thereby 
obtained in which the framing for the full cross-section may be 
gradually introduced. The framing is constructed with a cross- 



§ 171. 



TUNNELS. 



191 



section so large that the masonry lining may be constructed 
within it. 

171. Distinctive features of various methods of construction. 

There are six general systems, known as the English, German, 
Belgian, French, Austrian, and American. They are so named 




Fig. 94. 



from the origin of the methods, although their use is not con- 
fined to the countries named. Fig. 95 shows by numbers (1 to 5) 
the order of the excavation within the cross-sections. The Eng- 
lish, Austrian, and American s}'stems are alike in excavating the 
entire cross-section before beginning the construction of the 
masonry lining. The German method leaves a solid core (5) 
until practically the whole of the lining is complete. This has 
the disadvantage of extremely cramped quarters for work, poor 
ventilation, etc. The Belgian and French methods agree in 
excavating the upper part of the section, building the arch at 
otice, and supporting it temporarily until the side walls are 
built. The Belgian method then takes out the core (3), removes 
very short sections of the sides (4) immediately underpinning 
the arch with short sections of the side walls and thus gradually 
constructing the whole side wall. The French method digs out 
the sides (3), supporting the arch temporarily with timbers and 
then replacing the timbers with masonry; the core (4) is taken 
out last. The French method has the same disadvantage as the 
German— -working in a cramped space. The Belgian and French 
systems have the disadvantage that the arch, supported tempo- 
rarily on timber, is very apt to be strained and cracked by the 
slight settlement that so frequently occurs in soft material. The 
Enghsh, Austrian, and American methods differ mainly in thq 



192 



KAILROAD CONSTRUCTION. 



§ 171. 



design of the timbering„ The English support the roof by Knes 
of very heavy longitudinal timbers which are supported at com- 
paratively wide intervals by a heavy framework occupying the 



4 i 


1 


1 A 

1 


3 


1 
i 4 


+- 





_^| 


5 i 


1 


i 5 

! 




ENGLISH 



AUSTRIAN 





GERMAN 



BELGIAN 





FRENCH AMERICAN 

Fig. 95. — Order of Working by the Various Systems. 

whole cross-section. The Austrian system uses such frequent 
cross-frames of timber-work tha^ poling-boards will suffice to 
support the material between the frames. The American sys- 
tem agrees with the Austrian in using frequent cross-frames 



§ 172. TUNNELS. 193 

supporting poling-boards, but differs from it in that the ^' cross- 
frames" consist simply of arches of 3 to 15 wooden voussoirs, 
the voussoirs being blocks of 12''Xl2'' timber about 2 to 8 feet 
long and cut with joints normal to the arch. These arches are 
put together on a centering which is removed as soon as the arch 
is keyed up and thus immediately opens up the full cross-section, 
so that the center core (4) may be immediately dug out and the 
masonry constructed in a large open space. The American sys- 
tem has been used successfully in very soft ground, but its ad- 
vantages are greater in loose rock, when it is much cheaper than 
the other methods which emplov more timber. Fig. 90 and 
Plate III illustrate the use of the American system. Fig. 90 
shows the wooden arch in place. The masonry arch ma}' be 
placed when convenient, since it is possible to lay the track and 
commence traffic as soon as the wooden arch is in place. The 
student is referred to Drinker's "Tunneling" and to Eziha's 
^'Lehrbuch der Gesammten Tunnelbaukunst " for numerous 
illustrations of European methods of tunnel timbering. 

172. Ventilation during construction. Tunnels of any great 
length must be artificially ventilated during construction. If 
the excavated material is rock so that blasting is necessary, the 
need for ventilation becomes still more imperative. The inven- 
tion of compressed-air drills simultaneously solved two difilcul- 
ties. It introduced a motive power which is unobjectionable in 
its application (as gas would be), and it also furnished at the same 
time a supply of just what is needed — pure air. If no blasting 
is done (and sometimes even when there is blasting), air must be 
supplied by direct pumping. The cooling effect of the sudden 
expansion of compressed air only reduces the otherwise objection- 
ably high temperature sometimes found in tunnels. Since pure 
air is being continually pumped in, the foul air is thereby forced 
out. 

173. Excavation for the portals. Under normal conditions 
there is always a greater or less amount of open cut preceding 
and following a tunnel. Since all tunnel methods depend (to 
some slight degree at least) on the capacity of the exposed ma- 
terial to act as an arch, there is implied a considerable thickness 
of material above the tunnel. This thickness is reduced to 
nearly zero over the tunnel portals and therefore requires special 
treatment, particularly when the material is very soft. Fig. 96 * 

* Rziha, "Lehrbuch der Gesammten Tunnelbaukunst." 



194 



RAILROAD CONSTRUCTION, 



§ 174. 



illustrates one method of breaking into the ground at a portal. 
The loose stones are piled on the framing to giAe stability to the 
framing by their weight and also to retain the earth on the 




Fig. 96. — Timbering for Tunnel Portal. 



slope above. Another method is to sink a temporary shaft to 
the tunnel near the portal; immediately enlarge to the full size 
and build the masonry lining; then work back to the portal 
This method is more costly^ but is preferable in very treacherous 
ground, it being less liable to cause landslides of the surface 
material 

174. Tunnels vs. open cuts. In cases in which an open cut 
rather than a tunnel is a possibility the ultimate consideration 
is generally that of first cost combined with other financial con- 



§ 175. 



TUX>rELS. 



195 



siderations and annual maintenance charges directly or indirectly 
connected with it. Even Avhen an open cut may be constructed 
at the same cost as a tunnel (or perhaps a little cheaper) the 
tunnel may be preferable under the following conditions: 

1. When the soil indicates that the open cut would be liable 
to landslides. 

2. When the open cut would be subject to excessive snow- 
drifts or avalanches. 

3. When land is especially costly or it is desired to run under 
existing costly or valuable buildings or monuments. When run- 
ning through cities, tunnels are sometimes constructed as open 
cuts and then arched over. 

These cases apply to tunnels vs. open cuts when the align- 
ment is fixed by other considerations than the mere topography. 
The broader question of excavating tunnels to avoid excessive 
grades or to save distance or curvature, and similar problems, 
are hardly susceptible of general analysis except as questions of 
railway economics and must be treated individually. 

175. Cost of tunneling. The cost of any construction which 
involves such uncertainties as tunneling is very variable. It 
depends on the material encountered, the amount and kind of 
timbering required, on the size of the cross-section, on the price 
of labor, and especially on the reconstruction that may be neces- 
sary on account of mishaps. 

Headings generally cost $4 to $5 per cubic yard for excava- 
tion, while the remainder of the cross-section in the same tunnel 
may cost about half as much. The average cost of a large 
number of tunnels in this country may be seen from the follow- 
ing table : * 





{ 


C^ost per cubic yard 




Cost per 
lineal foot. 




Excavation. 


Masonry. 




Material. 


Single. 






Single. 


Double. 


Single. 


Double. 


Double. 


Hard rock 

Loose rock 

Soft ground. . . . 


$5.89 
3.12 
3.62 


$5.45 
3.48 
4.64 


$12.00 

9.07 

15.00 


$8.25 
10.41 
10.50 


$69 . 76 

80.61 

135.31 


$142.82 
119.26 
174.42 



* Figures derived from Drinker's "Tunneling.' 




(To face page 195.) 



LONGITDDINAI, SECTION OP PoKTAI. 



196 RAILROAD CONSTRUCTION. § 175. 

A considerable variation from these figures may be found in 
individual cases, due sometimes to unusual skill (or the lack of 
it) in prosecuting the work, but the figures will generally be 
sufficiently accurate for preliminary estimates or for the com- 
parison of two proposed routes. 



CHAPTER VI. 
CULVERTS AND MINOR BRIDGES. 

176. Definition and object. Although a variable percentage 
of the rain falling on any section of country soaks into the 
ground and does not immediately reappear, yet a very large 
percentage flows over the surface, always seeking and following 
the lowest channels. The roadbed of a railroad is constantly 
intersecting these channels, which frequently are normally dry. 
In order to pre^'ent injury to railroad embankments by the im- 
pounding of such rainfall, it is necessary to construct waterwa3's 
through the embankment through which such rainflow may 
freely pass. Such Avaterways, called culverts, are also appli- 
cable for the bridging of very small although perennial streams, 
and therefore in this work the term culvert will be applied to 
all water-channels passing through a railroad embankment 
which are not of sufficient magnitude to require a special struc- 
tural design, such as is necessary for a large masonry arch or a 
truss bridge. 

177. Elements of the design. A well-designed culvert must 
afford such free passage to the water that it will not ^^back up" 
OA^er the adjoining land nor cause any injury to the embankmeiit 
or culvert. The ability of the culvert to discharge freely all the 
water that comes to it evidently depends chiefly on the area of 
the waterway, but also on the form, length, slope, and materials 
of construction of the culvert and the nature of the approach 
and outfall. When the embankment is very low and the amount 
of water to be discharged very great, it sometimes becomes 
necessary to allow the water to discharge "under a head," i e., 
with the surface of the water above the top of the culvert. 
Safety then requires a much stronger construction than would 
othersvise be necessar}' to avoid injury to the culvert or embank- 
ment by washing. The necessity for such construction should 
be avoided if possible. 

197 



198 RAILROAD CONSTRUCTION. § 178. 



AREA OF THE WATERWAY. 

178. Elements involved. The determination of the required 
area of the waterway involves such a multipHcity of indeter- 
minate elements that any close determination of its value from 
purely theoretical considerations is a practical impossibility. 
The principal elements involved are: 

a. Rainfall. The real test of the culvert is its capacity to 
discharge without injury the flow resulting from the extraordi- 
nary rainfalls and ^' cloud bursts" that may occur once in many 
years. Therefore, while a knowledge of the average annual 
rainfall is of very little value, a record of the maximum rainfall 
during heavy storms for a long term of years may give a relative 
idea of the maximum demand on the culvert. 

b. Area of watershed. This signifies the total area of country 
draining into the channel considered. When the drainage area 
is very small it is sometimes included within the area surveyed 
by the preliminary survey. When larger it is frequently possi- 
ble to obtain its area from other maps with a percentage of 
accuracy sufficient for the purpose. Sometimes a special survey 
for the purpose is considered justifiable. 

c. Character of soil and vegetation. This has a large in- 
fluence on the rapidity with which the rainflow from a gi^'en 
area will reach the culvert. If the soil is hard and impermeable 
and the vegetation scant, a heav}^ rain will run off suddenly, 
taxing the capacity of the culvert for a short time, while a 
spongy soil and dense vegetation will retard the floAv, making it 
more nearly uniform and the maximum flow at any one time 
much less. 

d. Shape and slope of watershed. If the watershed is \'ery 
long and narrow (other things being equal), the water from the 
remoter parts will require so much longer time to reach the 
culvert that the flow will be comparatn'ely uniform, especially 
when the slope of the whole watershed is \ery low. When the 
slope of the remoter portions is quite steep it may result in the 
nearly simultaneous arri\ al of a storm- flow from all parts of the 
watershed, thus taxing the capacity of the culvert. 

e. Effect of design of culvert. The prmciples of h^drauhcs 
show that the slope of the culvert, its length, the form of the 
cross-section, the nature of the surface, and the form of the 



§ 179. CULVERTS AND IVIIXOR BRIDGES. 199 

approach and discharge all have a considerable influence on the 
area of cross-section required to discharge a given volume of 
water in a given time, but unfortunately the combined hy- 
draulic effect of these various details is still a very uncertain 
quantity. 

179. Methods of computation of area. There are three pos- 
sible methods of computation. 

(a) Theoretical. As shown a]:)Ove it is a practical impossi- 
bility to estimate correctly the combined effect of the great mul- 
tiplicity of elements which influence the final result. The nearest 
approach to it is to estimate by the use of empirical formula? 
the amount of water which will be presented at the upper end 
of the culvert in a given time and then to compute, from the 
principles of hydraulics, the rate of flow through a cuhert of 
given construction, but (as shown in § 178, e) such methods are 
still ^'ery unreliable, owing to lack of experimental kno^^ledge. 
This method has apparently greater scientific accuracy than 
other methods, but a little study will show that the elements 
of uncertainty are as great and the final result no more reliable. 
The method is most reliable for streams of uniform flow, but 
it is under these conditions that method (c) is most useful. The 
theoretical method v/ill not therefore be considered further. 

(bj Empirical. As illustrated in § 180, some formulae make 
the area of waterway a function of the drainage area, the for- 
mula bemg affected by a coefficient the value of which is esti- 
mated between hmits according to the judgment Assummg 
that the formula? are sound, their use onl}' narrows the limits of 
error, the final determination depending on experience and 
judgment. 

(c) From observation. This method, considered by far the 
best for permanent work, consists m observing the high-water 
marks on contracted channel-openings which are on the same 
stream and as near as possible to the proposed culvert. If the 
country is new and there are no such openings, the wisest plan 
is to bridge the opening by a temporary structure in wood which 
has an ample waterway (see § 126, h, 4) and carefully observe 
all high- water marks on that opening during the 6 to 10 years 
which is ordinarily the minimum life of such a structure. As 
shown later, such observations may be utilized for a close com- 
putation of the required waterway. Method (b) may be utilized 
for an approximate calculation for the reoui^ed area for the tem- 



233 KAILROAD CONSTRUCTION. § ISO, 

porary structure, using a value which is intentionally excessive, 
so that a permanent structure of sufficient capacit}^ may subse- 
quently be constructed within the temporary structure. 

1 80. Empirical formulae. Two of the best known empirical 
formulae for area of the waterway are the following: 

(a) Myer's formula: 

Area of waterway in square feet = (7 X \/drainage area in acres, 
where (7 is a coefficient varying from 1 for flat country to 4 for 
mountainous country and rocky ground. As an illustration, if 
the drainage area is 100 acres, the waterway area should be from 
10 to 40 square feet, according to the value of the coefficient 
chosen. It should be noted that this formula does not regard 
the great variations in rainfall in various parts of the world nor 
the design of the culvert, and also that the final result depends 
largely on the choice of the coefficient. 

(b) Talbot's formula: 

Area of waterway in square feet ==CX\/(drainage area in acres) ^. 
'* For steep and rocky ground C varies from § to 1. For rolling 
agricultural country subject to floods at times of melting snow, 
and Avith the length of the valley three or four times its width, C 
is about i-; and if the stream is longer in proportion to the area, 
decrease C. In districts not affected by accumulated snow, and 
where the length of the valley is several times the width, ^ or J , 
or even less, may be used. C should be increased for steep side 
slopes, especially if the upper part of the valley has a much 
greater fall than the channel at the culvert." * As an illus- 
tration, if the drainage area is 100 acres the area of waterway 
should be CX31.6. The area should then vary from 5 to 31 
square feet, according to the character of the country. Like 
the previous estimate, the result depends on the choice of a 
coefficient and disregards local variations in rainfall, except as 
they may be arbitrarily allowed for in choosing the coeffi- 
cient. 

181. Value of empirical formulae. The fact that these for- 
mulae, as well as many others of similar nature that have been 
suggested, depend so largely upon the choice of the coefficient 
shows that they are valuable '' more as a guide to the judgment 
than as a working rule," as Prof. Talbot explicitly declares in 



* Prof. A. N. Talbot, "Selected Papers of the Civil Engineers' Club of 
the Univ. of Illinois." 



§ 182. CULVERTS AND MINOR BRIDGES. 201 

commenting on his own formula. In short, they are chiefly valu- 
able in indicating a probable maximum and minimum between 
which the true result probably lies. 

182. Results based on Observation. As already indicated in 
§ 179, observation of the stream in question gives the most 
rehable results. If the country is new and no records of the 
flow of the stream during heavy storms has been taken, even 
the life of a temporary wooden structure may not be long enough 
to include one of the unusually severe storms which must be 
allowed for, but there will usually be some high-water mark 
which will indicate how much opening will be required. The 
following quotation illustrates this: "A tidal estuary may gen- 
erally be safely narrowed considerably from the extreme water 
lines^ if stone revetments are used to protect the bank from 
wash. Above the true estuary, where the stream cuts through 
the marsh, we generally find nearly vertical banks, and we are 
safe if the faces of abutments are placed even with the banks. 
In level sections of the country, where the current is sluggish, 
it is usually safe to encroach somewhat on the general width 
of the stream, but in rapid streams among the hills the width 
that the stream has cut for itself through the soil should not be 
lessened, and in ravines carrying mountain torrents the open- 
ings must be left very much larger than the ordinary appear- 
ance of the banks of the stream would seem to make neces- 

sarv." * 

As an illustration of an observation of a storm-flow through 
a temporary trestle, the following is quoted: " Having the flood 
height and velocity, it is an easy matter to determine the vol- 
ume of water to be taken care of. I have one ten-bent pile 
trestle 135 feet long and 24 feet high over a spring branch that 
ordinarily runs about six cubic inches per second. Last sum- 
mer during one of our heavy rainstorms (four inches in less 
than three hours) I visited this place and found by float obser- 
vations the surface velocity at the highest stage to be 1.9 feet 
per second. I made a high-watermark, and after the flood- 
water receded found the width of stream to be 12 feet and an 
average depth of 2| feet. This, with a surface velocity of 1.9 
feet per second, would give approximately a discharge of 50 



* J. P. Snow, Boston & Maine Railway. From Report to Association of 
Railway Superintendents of Bridges and Buildings. 1897. 



202 RAILROAD CONSTRUCTION. § 183. 

cubic feet, or 375 gallons, per second. ^Having this information 
it is easy to determine size of opening required." * 

183. Degree of accuracy required. The advantages result- 
ing from the use of standard designs for culverts (as well as 
other structures) have led to the adoption of a comparatively 
small number of designs. The practical use made of a compu- 
tation of required waterway area is to determine which one of 
several standard designs will most nearly fulfill the require- 
ments. For example, if a 24-inch iron pipe, having an area of 
3.14 square feet, is considered to be a little small, the next size 
(30-inch) would be adopted; but a 30-inch pipe has an area of 
4.92 square feet, which is 56% larger. A similar result, except 
that the percentage of difference might not be quite so marked, 
will be found by comparing the areas of consecutive standard 
designs for stone box culverts. 

The advisability of designing a culvert to withstand any 
storm-flow that may ever occur is considered doubtful. Several 
years ago a record-breaking storm in New England carried 
away a very large number of bridges, etc., hitherto supposed 
to be safe. It was not afterward considered that the design of 
those bridges was faulty, because tke extra cost of constructing 
bridges capable of withstanding such a flood, added to interest 
for a long period of years, would be enormously greater than the 
cost of repairing the damages of such a storm once or twice in 
a century. Of course the element of danger has some weight, 
but not enough to justify a great additional expenditure, for 
common prudence would prompt unusual precautions during 
or immediately after such an extraordinary storm. 

PIPE CULVERTS. 

184. Advantages. Pipe culverts, made of cast iron or earthen- 
ware, are very durable, readily constructed, moderately cheap, 
will pass a larger volume of water in proportion to the area than 
many other designs on account of the smoothness of the sur- 
face, and (when using iron pipe) may be used very close to 
the track when a low opening of large capacity is required. 
Another advantage lies in the ease with which they may be 
inserted through a somewhat larger opening that has been 

* A. J. Kelley, Kansas City Belt Railway. From Report to Association 
of Railway Superintendents of Bridges and Buildings. 1897. 



§ 185. CULVERTS AND MIXOR BRIDGES. 203 

temporarily lined with wood, without disturbing the roadbed 
or track. 

185. Construction. Permanency requires that the founda- 
tion shall be firm and secure against being washed out. To 
accomplish this, the soil of the trench should be hollowed out to 
fit the lower half of the pipe, making suitable recesses for the 
bells. In very soft treacherous soil a foundation-block of con- 
crete is sometimes placed under each joint, or even throughout 
the whole length. When pipes are laid through a slightly 
larger timber culvert great care should be taken that the pipes 
are properly supported, so that there will be no settling nor 
development of unusual strains when the timber finally decays 
and gives way. To prevent the washing away of material 
around the pipe the ends should be protected by a bulkhead. 
This is best constructed of masonry (see Fig. 97), although wood 
is sometimes used for cheap and minor constructions. The joints 
should be calked, especially when the culvert is liable to run 
full or when the outflow is impeded and the culvert is liable to 
be partly or wholly filled during freezing weather. The cost of 
a calking of clay or even hydraulic cement is insignificant com- 
pared with the value of the additional safet}' afforded. When 
the grade of the pipe is perfectly uniform, a very low rate of 
grade will suffice to drain a pipe culvert, but since some uneven- 
ness of grade is inequitable through uneven settlement or im- 
perfect construction, a grade of 1 in 20 should preferably be 
required, although much less is often used. The length of a 
pipe culvert is approximately determined as follows: 

Length = 2s (depth of emhankment to top of pipe) + (width of roadbed), 

in which s is the slope ratio (horizontal to vertical) of the banks. 
In practice an even number of lengths will be used which will 
most nearly agree with this formula. 

186. Iron-pipe culverts. Simple cast-iron pipes are used in 
sizes from 12'' to 48" diameter. These are usually made in 
lengths of 12 feet with a few lengths of 6 feet, so that am^ required 
length may be more nearly obtained. The lightest pipes made 
are sufficiently strong for the purpose, and even those which 
would l)e rejected because of incapacity to withstand pressure 
may be utilized for this work. In Fig. 97 are shown the stand- 
ard plans used on the C. C. C. & St. L. Ry., which may be con- 
sidered as typical plans. 










\ 




' ^ 






\ 










\ 














^o 






\ 
















<M 






\ 






















CM 


\ 


















/<^^^--T"~~~^ 








Q 






//T^^^ 








+ 




^0 


z\m a 




,6 NVHjL A 


*• 


ss3n xoN -1 


00 






\\\ /h 






















t- 






\n>^.^-^ 




















r7 










































t 












\ 






/ 
















v'f 






/ 
















cvr 






/ 






















/ 


/ 





k-T^/i 



-a -HO/ 



Fig. 97. — Standard Cast-iron 
Pipe Culvert. C. C. C. & 
St. L. Ry. (May 1893.) 



§ 187. 



CULVERTS AND MINOR BRIDGES. 



205 



Pipes formed of cast-iron segments have been used up to 12 
feet diameter. The shell is then made comparatively thin, but 
is stiffened by ribs and flanges on the outside. The segments 
break joints and are bolted together thi'ough the flanges. The 
joints are made tight by the use of a tarred rope, together with 
neat cement. 

187. Tile- pipe culverts. The pipes used for this purpose 
vary from 12'' to 24'' in diameter. When a larger capacity is 
required two or more pipes may be laid side by side, but in 
such a case another design might be preferable. It is frequently 
specified that '^double-strength" or '' extra-heavy'' pipe shall 
be used, evidently with the idea that the stresses on a culvert- 
pipe are greater than on a sewer-pipe. But it has been con- 
clusively demonstrated that, no matter how deep the embank- 
ment, the pressure cannot exceed a somewhat unceitain maxi- 
mum, also that the greatest danger consists in placing the pipe 
so near the ties that shocks may be directly transferred to the 
pipe T\dthout the cushioning effect of the earth and ballast. 
When the pipes are well bedded in clear earth and there is a 




UP-STREAr/,^ETsiD. 



DOWN-STREAM E'ND. 



DOWN-STREAM END. THREE PIPES* 



Fig. 98. — Standard Vitrified-pipe Culvert. Plant System. (189.1.) 

sufficient depth of earth over them to avoid direct impact (at 
least three feet) the ordinary sewer-pipe will be sufficiently 
strong. ''Double-strength" pipe is fi'equently less perfectly 
burned, and the supposed extra strength is not therefore ob- 



208 



RAILROAD CONSTRUCTION. 



§18S. 



tained. In Fig. 98 are shown the standard plans for vitrified- 
pipe culverts as used on the ^' Plant system." Tile pipe is much 
cheaper than iron pipe, but is made in much shorter lengths and 
requires much more work in laying and especially to obtain a 
uniform grade. 

BOX CULVERTS. 

1 88. Wooden box culverts. This form serves the purpose 
of a cheap temporary construction which allows the use of a 
ballasted roadbed. As in all temporary constructions, the area 
should be made considerably larger than the calculated area 
§§ 179-182), not only for safety but also in order that, if the 
smaller area is demonstrated to be sufficiently large, the per- 
manent construction (probably pipe) may be placed inside with- 
out disturbing the embankment. All designs agree in using 
heavy timbers (12"Xl2'', 10''Xl2", or 8''Xl2'0 for the side 
walls, cross-timbers for the roof, every fifth or sixth timber 
being notched down so as to take up the thrust of the side walls, 
and planks for the flooring. Fig. 99 shows some of the standard 
designs as us^d b}^ the C, M. & St. P. Ry. 



W^TTT^y 







V\G. 99. — Standard Timber Box Culvert. C, M. & St. P. Ry. 

(Feb. 1889.) 

189. Stone box culverts. In localities where a good quality 
of stone is cheap, stone box culverts are the cheapest form of 
permanent construction for culverts of medium capacity, but 
their use is decreasing owing to the frequent difficulty in obtain- 
ing really suitable stone within a reasonable distance of the 
culvert. The clear span of the cover-stones varies from 2 to 4 
feet. The required thickness of the cover-stones is sometimes 



§ 189. 



CULVERTS AND MINOR BRIDGES. 



207 



calculated by the theory of transverse strains on the basis of 
certain assumptions of loading — as a function of the height of 
the embankment and the unit strength of the stone used. Such 
a method is simply another illustration of a class of calculations 
which look very precise and beautiful, but which are worse than 
useless (because misleading) on account of the hopeless uncer- 





LONG. SEC. 



PLAN 

Fig. 100.— Standard Single Stone Culvert (3'X4'). N. & W. R.R. 

(1890.) 

tainty as to the true value of certain quantities which must be 
used in the computations In the first place the true value of 
the unit tensile strength of stone is such an uncertain and variable 



208 



RAILROAD GONSTRUCTION. 



§189, 



quantit}^ that calculations based on any assumed value for it are 
of small reliability. In the second place the weight of the prism 
of earth lying directly above the stone, plus an allowance for live 
load, is by no means a measure of the load on the stone nor of 
the forces that tend to fracture it. All earthwork will tend to 




t^5^ 




^ 



COVER 



STONES 




PLAN 



Fig. 100a. — Standard Double Stone Culvert (3'X40. N. & W. R. R. 

(1890.) 

form an arch aboA'e any cavity and thus relieve an uncertain 
and probably variable proportion of the pressure that might 
otherwise exist. The higher the embankment the less the pro- 



§ 190, 



CULVERTS AND MINOR BRIDGES. 



209 



portionate loading, until at some uncertain height an increase 
in height will not increase the load on the cover-stones. The 
effect of frost is like^^ ise large, but uncertain and not computable. 
The usual practice is therefore to make the thickness such as 
experience has shown to be safe with a good quahty of stone, 
i.e., about 10 or 12 inches for 2 feet span and up to 16 or 18 
inches for 4 feet span. The side walls should be carried down 
deep enough to prevent their being undermined by scour or 
heaved by frost. The use of cement mortar is also an important 
feature of first-class work, especially when there is a rapid scour- 
ing current or a liability that the culvert will run under a head. 
In Figs. 100 and 100a are shown standard plans for single and 
double^stone box culverts as used on the Xorfolk'and Vrestern R R. 
190. Old-rail culverts. It sometimes happens (although very 
rarely) that it is necessary to bring the grade line within 3 or 4 
feet of the bottom of a stream and yet allow an area of 10 or 12 
square feet. A single large pipe of sufficient area could not be 
used in this case. The use of several smaller pipes side by side 
would be both expensive and inefficient. For similar reasons 
neither wooden nor stone box culverts could be used. In such 
cases, as well as in many others where the head-room is not so 
limited, the plan illustrated in Fig. 101 is a very satisfactory 




Fig. 101.— Standard Old-rail Culvert. N. & W. R.R. (1895.) 



solution of the problem. The old rails, having a length of 8 or 
9 feet, are laid close together across a 6-foot opening. Some- 
times the rails are held together by long bolts passing through 



J 



210 



RAILROAD CONSTRUCTION. 



§ 191. 



the webs of the rails. In the plan shown the rails are confined 
by low end walls on each abutment. This plan requires only 15 
inches between the base of the rail and the top of the culvert 
channel. It also gives a continuous ballasted roadbed. 



ARCH CULVERTS. 

191. Influence of design on flow. The variations in the design 
of arch culverts have a very marked influence on the cost and 
efficiency. To combine the least cost with the greatest effi- 
ciency, due weight should be given to the following elements: 
(a) amount of masonry, (b) the simplicity of the constructive 
work, (c) the design of the wing walls, (d) the design of the 
junction of the wing walls with the barrel and faces of the arch, 
and (e) the safety and permanency of the construction. These 
elements are more or less antagonistic to each other, and the 
defects of most designs are due to a lack of proper proportion 
in the design of these opposing interests. The simplest con- 
struction (satisf}'ing elements h and e) is the straight barrel arch 
between tAvo parallel vertical head walls, as sketched in Fig. 
102, a. From a h3^draulic standpoint the design is poor, as the 
water eddies around the corners, causing a great resistance 
which decreases the flow. Fig. 102, h, shows a much better de- 




FiG. 102. — Types of Culverts. 



sign in many respects, but much depends on the details of the 
design as indicated in elements (h) and (d). As a general thing 
a good hydraulic design requires complicated and expensive 
masonry construction, i.e., elements (h) and (d) are opposed. 
Design 102, c, is sometimes inapplicable because the Avater is 



J L 



STANDARD ARCH 

8 FEET SPA 

NORFOLK & WES 
(1891) 



^ 




HALF END 



HALF SECTIOI 



note:- in plage of brick arch, 
rubble stone arch of same thick- 
mess may be used 



(To jace page 211.) 



STANDARD ARCH CULVERT 

8 FEET SPAN 

NORFOLK & WESTERN R.R. 
(1891) 




tl 

I 



192 



CULVERTS AND MINOR BRIDGES, 



211 



liable to ^Aork in behind the masonry during floods and perhaps 
cause scour. This design uses less masonry than (a) or (b). 

192. Example of arch culvert design. In Plate IV is shown 
the design for an 8-foot arch culvert according to the standard 
of the Norfolk and Western R. R. Note that the plan uses the 
flaring wing walls (Fig. 102, h) on the up-stream side (thus 
protecting the abutments from scour) and straight wing walla 
(similar to Fig. 102, c) on the down-stream end. This econo- 
mizes masonr}-^ and also simplifies the constructive work. Note 
also the simplicity of the junction of the wing walls with the 
barrel of the arch, there being no re-entrant angles below the 
springing line of the arch. The design here shown is but one 
of a set of designs for arches varying in span from 6' to 30'. 



MINOR OPENINGS. 

193. Cattle-guards, (a) Pit guards. Cattle-guards will be 
considered under the head of minor openings, since the old- 



4 X 13 X 7 



12x12x138" 



•11x14x6 




12 10- 



■6x12" 



Fig. 103.— Pit Cattle-guards. P. R.R. 



fashioned plan of pit guards, which are even now defended and 
preferred by some railroad men, requires a break in the con- 
tinuity of the roadbed. A pit about three feet deep, five feet 
long, and as wdde as the width of the roadbed, is walled up with 



212 



RAILROAD CONSTRUCTION, 



§ 193, 



stone (sometimes with wood), and the rails are supported on 
heavy timbers laid longitudinally with the rails. The break in 
the continuity of the roadbed produces a disturbance in the 
elastic wave running through the rails, the effect of which is 
noticeable at high velocities. The greatest objection, however, 
lies in the dangerous consequences of a derailment or a failure 
of the timbers owing to unobserved decay or destruction by 
fire — caused perhaps by sparks and cinders from passing loco- 
motives. The very insignificance of the structure often leads 
to careless inspection. But if a single pair of wheels gets off 
the rails and drops into the pit, a costly wreck is inevitable. 
The (once) standard design for such a structure on the Penn- 
sylvania R.R. is shown in Fig. 103. 

(b) Surface cattle-guards. These are fastened on top of the 
ties; the continuity of the roadbed is absolutely unbroken and 




Fig. 104. — =Cattle-guard with Wooden Slats. 

thus is avoided much of the danger of a bad wreck owing to a 
possible derailment. The device consists essentially of overlay- 
ing the ties (both inside and outside the rails) with a surface on 
which cattle will not walk. The multitudinous designs for such 
a surface are variously effective in this respect. An objection, 



§ 194. 



CULVERTS AND MINOR BRIDGES. 



213 



which is often urged indiscriminately against all such designs, is 
the liabihty that a brake-chain which may happen to be drag- 
ging may catch in the rough bars which are used. The bars 
are sometimes "home-made/' of wood, as shown in Fig. 104. 
Iron or steel bars are made as shown in Fig. 105. The general 
construction is the same as for the wooden bars. The metal 
bars have far greater durabhity, and it is claimed that they are 
more effective in discouraging cattle from attempting to cross. 




Fig. 105. — Merrill-Stevens Steel Cattle-guard. 

194. Cattle-passes. Frequently when a railroad crosses a 
farm on an embankment, cutting the farm into two parts, the 
railroad company is obliged to agree to make a passageway 
through the embankment sufficient for the passage of cattle and 
perhaps even farm-wagons. If the embankment is high enough 
so that a stone arch is practicable, the initial cost is the only 
great objection to such a construction; but if an open wooden 
structure is necessary, all the objections against the old-fashioned 
cattle-guards apply with equal force here. The avoidance of a 
grade crossing which Avould otherwise be necessary is one of the 
great compensations for the expense of the construction and 
maintenance of these structures. The construction is some- 
times made by placing two pile trestle bents about 6 to 8 feet 
apart, supporting the rails by stringers in the usual way, the 
special feature of this construction being that the embankments 
are filled in behind the trestle bents, and the thrust of the em- 
bankments is mutually taken up through the stringers, which 
are notched at the ends or otherAvise constructed so that they 
may take up such a thrust. The designs for old-rail culverts 
and arch culverts are also utihzed for cattle-passes when suitable 
and convenient, as well as the designs illustrated in the following 
section. 

195. Standard stringer and I-beam bridges. The advantages 
of standard designs apply even to the covering of short spans 



± 



214 KAILROAD CONSTRUCTION. § 195. 

with wooden stringers or with I beams — especially since the 
methods do not require much vertical space between the rails 
and the upper side of the clear opening, a feature which is often 
of prime importance. These designs are chiefly used for cul- 
verts or cattle-passes and for crossing over highways — proA^ding 
such a narrow opening would be tolerated. The plans all imply 
stone abutments, or at least abutments of sufficient stability to 
withstand all thrust of the embankments. Some of the designs 
are illustrated in Plate V. The preparation of these standard 
designs should be attacked by the same general methods as 
already illustrated in § 156. When computing the required 
transverse strength, due allowance should be made for lateral 
bracing, which should be amply provided for. Note particu- 
larly the methods of bracing illustrated in Plate V. The designs 
calling for iron (or steel) stringers may be classed as permanent 
constructions, which are cheap, safe, easily inspected and main- 
tained, and therefore a desirable method of construction. 



f 






n 




TYPE "E" GIRDER 




-tiu^-^^^^ST^ 




STANDARD I-BRIDGES-14-FT. SPAN. 

NORFOLK AND WESTERN R.R. 



(■/'o /ace page 214.) 



^ 



CHAPTER Vn. 
BALLAST. 

196. Purpose and requirements. "The object of the ballast 
is to transfer the applied load over a large surface; to hold the 
timber work in place horizontally; to carry off the rain-water 
from the superstructure and to prevent freezing up in winter; 
to afford means of keeping the ties truh' up to the grade line; 
and to give elasticity to the roadbed. '^ This extremely con- 
densed statement is a description of an ideally perfect ballast. 
The value of any given kind of ballast is proportional to the 
extent to which it fulfills these requirements. The ideally 
perfect ballast is not necessarily the most economical ballast 
for all roads. Light traffic generally justifies something cheaper, 
but a very common error is to use a very cheap ballast when a 
small additional expenditure would procure a much better 
ballast which would be much more economical in the long run. 

197. Materials. The materials most commonly employed are 
gravel and broken stone. Burnt clay, cinders, shells, and small 
coal are occasionally used as ballast when they are especially 
cheap and convenient or when better kinds are especially expen- 
sive. Although it is hardly correct to speak of the natural soil 
as ballast, yet many miles of cheap railways are "ballasted'^ 
with the natural soil, which is then called " mud ballast.'' 

Mud ballast. When the natural soil is gravelly so that rain 
win drain through it quickly, it will make a fair roadbed for 
light traffic, but for heavy traffic, and for the greater part of 
the length of most roads, the natural soil is a ver^^ poor material 
for ballast; for, no matter how suitable the soil might be along 
limited sections of the road, it would practically never happen 
that the soil would be uniformly good throughout the whole 
length of the road. Considering that a heavy rain will in one 
day spoil the results of weeks of patient "surfacing" mth mud 
ballast, it is seldom economical to use "mud" if there is a 

315 



216 RAILROAD CONSTRUCTION. § 197. 

gravel -bed or other source of ballast anywhere on the line of 
the road 

Cinders. The advantages consist in the excellent facilities 
for drainage, ease of handling, and cheapness — after the road is 
in operation One disadvantage is excessive dust in dry weather 
Cinders are considered preferable to grav^el in yards. 

Slag. When slag is readily obtainable it furnishes an ex- 
cellent ballast, free from dust and perfect in drainage qualities 
Some kinds of slag are objectionable on account of their delete- 
rious chemical effect on the ties and spikes — especially on 
metalhc ties. 

Shells, small coal, etc. These comparatively inferior kinds 
of ballast are used for light traffic when they are especially cheap 
and conA^enient. They are extremely dusty in dry weather, 
break up into very fine dust, and are but little better than 
mud. 

Gravel. This is the most common form of ballast which may 
be called good ballast. In 1885, the Roadmasters Association 
of America voted in favor of gravel ballast as against rock bal- 
last. Although not so stated, this action was perhaps due to a 
conviction of its real economy for the average railroad of this 
country, which may be called a ^' light traffic '^ road. Gravel 
should preferably be screened over a screen ha^dng a y mesh, 
so as to screen out all dirt and the finest stones. Generally a 
railroad will be able to find at some point along its line a '' gravel- 
pit" affording a suitable supply. This may be dug out w^th a 
steam-shovel, screened if necessary, and sent out over the line 
by the train-load at a comparatively small cost. 

Rock or broken stone. Rock ballast is generally specified to 
be such as will pass through a IV' (or 2'0 ring. Although 
preferably broken by hand, machine-broken stone is much 
cheaper. It is most easily handled with forks. This also has 
the effect of screening out the dirt and fine chips which would 
interfere with effectual drainage. Rock ballast is more expen- 
sive in first cost, and also more troublesome to handle, than any 
other kind, but under heavy traffic will keep in surface better 
and will require less work for maintainance after the ties have 
become thoroughly bedded. For roads with very hght traffic, 
running few trains, at comparatively low velocities, the advan- 
tages of rock ballast over other kinds are not so pronounced. 
For such roads rock ballast is an expensive luxury. The amount 



§ 198. 



BALLAST. 



217 



of traffic which will justify the ut?e of rock ballast wiJl depend 
on the cost of obtaining ballast of the various knids. 

198. Cross-seciions. A depth of 12^' under the tie is gener 
ally required on the best roads, but for light traffic this is some- 
times reduced to 6'' and even less. The a\ idth is generally 1 to 
2 feet less than the width of the roadbed pioper — excluding 
ditches. If the ballast has an average width of 10 feet (12 feet 
at bottom and 8 feet ai top) and an average depth of 15 inches 
(including that placed between the ties), it will lequire 2444 
cubic 3^ards per mile of track. The P R. R. estimates 2500 
cubic yards of gravel and 2800 cable yards of stone ballast per 
mile of single track. On account of the lequirements of drain- 
age the best form of cross- section depends on the kind of ballast 
used. 

Mud ballast. Since the great objection to mud ballast lies in 
its liability to become soft by soaking up the rain that falls, it 
becomes necessary that it should be drained as quickly and 
readily as its nature will permit Fig. 106 shows a typical cross- 




FiG. 106. — "Mud" Ballast. 

section for mud ballast It should be croAvned 2" above the 
top of the tie at the center, thence sloped so as to leave a slight 
clearance under the rail between the ties^ thence sloping doA^n 
to the bottom of the tie at each end and continuing to slope 
down to the ditch (in cut), w^hich should be 18' or 20'' below 
the bottom of the tie. 

Gravel, cinders, slag, etc. The subgrade is crowned 6'*' or 
8'' in the center^ as shoAvn in Fig. 107 The ballast is crowned 




,FiG. 107. — Gravel Ballast. 

to the top of the tie in the center, but is sloped down to the 
bottom of the tie at each end This is necessary (and more 



218 RAILROAD CONSTRUCTION. § 199. 

especially so with mud ballast) to prevent a possible accumula- 
tion and settlement of water at the ends of the tie, which would 
readily soak into the end fibers and produce decay. 

Broken stone. Stone ballast is shouldered out beyond the 
ends of the ties so as to afford greater lateral binding. The 
space between the ties is filled up level with the tops. The 




Fig. 108. — Broken Stone Ballast. 

perfect drainage of stone ballast permits this to be done without 
any danger of causing decay of the tiesi by the accumulation 
and retention of water. 

199. Methods of laying ballast. The cheapest method of 
laying ballast on new roads is to lay ties and rails directly on 
the prepared subgrade and run a construction train over the 
track to distribute the ballast. Then the track is lifted up until 
sufficient ballast is worked under the ties and the track is prop- 
erly surfaced. This method, although cheap, is apt to injure 
the rails by causing bends and kinks, due to the passage of 
loaded construction trains when the ties are very unevenly and 
roughly supported, and the method is therefore condemned and 
prohibited in some specifications The best method is to draw 
in carts (or on a contractor's temporary track) the ballast that is 
required under the level of the bottom of the ties. Spread this 
ballast carefully to the required surface Then lay the ties and 
rails, which will then have a very fair surface and uniform sup- 
port. A construction train can then be run on the rails and 
distribute sufficient additional ballast to pack around and 
between the ties and make the required cross-section 

The necessity for constructing some lines at an absolute mini- 
mum of cost and of opening them for traffic as soon as possible 
has often led to the policy of starting traffic when there is little 
or no ballast — perhaps nothing more than a mere tamping of 
the natural soil under the ties. When this is done ballast may 
subsequently be drawn where required by the train-load on 
flat cars and unloaded at a minimum of cost by means of a 
'^plough '^ The plough ha« the same width as the cars and is 



§ 200. BALLAST. 219 

guided either by a ridge along the center of each car or by shoit 
posts set up at the sides of the cars It is drawn from one end 
of the train to the other by means of a cable. The cable is 
sometimes operated by means of a small hoisting-engine carried 
on a car at one end of the train. Sometimes the locomotive is 
detached temporarily from the train and is run ahead with the 
cable attached to it. 

200. Cost. The cost of baJlast in the track is quite a variable 
item for different roads, since it depends (a) on the first cost of 
the material as it comes to the road, (b) on the distance from 
the source of supply to the place where it is used, and (c) on 
the method of handling. The first cost of cinder or slag is 
frequently insignificant A gravel-pit ma\^ cost nothing except 
the price of a little additional land be^'ond the usual limits of the 
right of way. Broken stone will usually cost $1 or more per 
cubic yard If suitable stone is obtainable on the company's 
land, the cost of blasting and breaking should be somewhat less 
than this The cost of loading the ballast on to trains wdll be 
small (per cubic yard) if it is handled with steam-shovels — as in 
the case of gravel taken from a gravel-pit Hand-shovelling 
will cost more. The cost of hauling will depend on the distance 
hauled, and also^ to a considerable extent^ on the limitations on 
the operation of the train due to the necessity of keeping out of 
the way of regular trains There is often a needless waste in 
this way. The ^'mad train" is considered a pariah and entitled 
to no rights whatever, regardless of the large daily cost of such 
a train and of the necessary gang of men. The cost of broken 
stone ballast in the track is estimated at $1 25 per cubic yard. 
The cost of gravel ballast is estimated at GO c. per cubic yard 
in the track. The cost of placing and tamj)ing gravel ballast is 
estimated at 20 c. to 24 c. per cubic yard, for cinders 12 c. to 
15 c. per cubic yard. The cost of loading gravel on cars, using 
a steam-shovel, is estimated at 6 c. to 10 c. per cubic yard.* 

* Report Roadmasters Association. 1885. 



CHAPTER VIII. 

TIES, 
AND OTHER FORMS OF RAIL SUPPORT. 

201, Various methods of supporting rails. It is necessary 
that the rails shall be sufficienth' supported and braced, so that 
the gauge shall be kept constant and that the rails shall not be 
subjected to excessive transverse stress. It is also preferable 
that the rail support shall be neither rigid (as if on solid rock) 
nor too 3delding, but shall have a uniform elasticity throughout. 
These requirements are more or less fulfilled by the following 
methods. 

(a) Longitudinals. Supporting the rails throughout their 
entire length. This method is very seldom used in this country 
except occasionally on bridges and in terminals when the 
longitudinals are supported on cross-ties. In § 224 will be 
described a system of rails, used to some extent in Europe, 
having such broad bases that they are self-supporting on the 
ballast and are only connected by tie-rods to maintain the 
gauge. 

(b) Cast-iron "bowls" or "pots." These are castings resem- 
bling large inverted bowls or pots, having suitable chairs on 
top for holding and supporting the rails, and tied together wdth 
tie-rods. They will be described more fully later (§ 223). 

(c) Cross-ties of metal or wood. These will be discussed in 
the following sections, 

202. Economics of ties. The true cost of ties depends on the 
relative total cost of maintenance for long periods of time. The 
first cost of the ties delivered to the road is but one item in the 
economics of the question. Cheap ties require frequent renew- 
als, which cost for the labor of each renewal practically the 
same whether the tie is of oak or of hemlock. Cheap ties make 
a poor roadbed which will require more track labor to keep even 
in tolerable condition. The roadbed will require to be disturbed 
so frequently on account of renewals that the ties never get an 

220 



§ 203. TIES. 221 

Oi3port unity to get settled and to form a smooth roadbed for any 
length of time. Irregularity in width, thickness, or length of 
ties is especially detrimental in causing the ballast to act and 
wear unevenl}^ The life of ties has thus a more or less direct 
influence on the life of the rails, on the wear of rolling stock, and 
on the speed of trains. These last items are not so readily 
reducible to dollars and cents, but ^vhen it can be shown that 
the total cost, for a long period of time, of several renewals of 
cheap ties, with all the extra track labor involved, is as great as 
or greater than that of a few renewals of durable ties, then there 
is no question as to the real economy. In the following dis- 
cussions of the merits of untreated ties (either cheap or costly), 
chemically treated ties, or metal ties, the true question is there- 
fore of the ultimate cost of maintaining any particular kind of 
ties for an indefinite period, the cost including the first cost of 
the ties, the labor of placing them and m.aintaining them to 
surface, and the somewhat uncertain (but not therefore non- 
existent) effect of frequent renewals on repairs of rolling stock, 
on possible speed, etc. 



W^OODEN TIES. 

203. Choice of wood. This naturally depends, for any partic- 
ular section of country, on the supply of wood v»'hich is most 
readily available. The woods most commonl^^ used, especially 
in this countr}^, are oak and pine, oak being the most durable 
and generally the most expensive. Redwood is used very ex- 
tensively in California and proves to be extremely durable, so 
far as decay is concerned, but it is very soft and is much injured 
by ^'rail-cutting." This defect is being partly remedied by the 
use of tie-plates, as T\'ill be explained later. Cedar, chestnut, 
hemlock, and tamarack are frequently used in this country. In 
tropical countries very durable ties are frequently obtained from 
the hard woods peculiar to those countries. According to a 
bulletin of the U. S. Department of Agriculture issued some 
3'ears ago, the proportions of the various kinds usied in the 
United States are about as follows: 



Oak 605 

Pine 20 

Cedar 6 



Chestnut 5% 

Hemlock and Ta- 
marack 3 

Redwood 3 



Cypress 2% 

Various 1 

Total 100^ 



222 RAILROAD CONSTRUCTION. § 204. 

The limitations of timber supply have somewhat dimin- 
ished the use of oak and increased the use of the softer woods 
in recent years. 

204. Durability. The durability of ties depends on the cli- 
mate; the drainage of the ballast; the volume, w^eight, and 
speed of the traffic; the curvature, if an}-; the use of tie-plates; 
the time of year of cutting the timber; the age of the timber 
and the degree of its seasoning before placing in the track; the 
nature of the soil in which the timber is grown; and, chiefly, 
on the species of wood employed. The variability in these 
items will account for the discrepancies in the reports on the life 
of various woods used for ties. 

White oak is credited with a life of 5 to 12 3^ears, depending 
principally on the traffic. It is both hard and durable, the 
hardness enabling it to withstand the cutting tendency of the 
rail-fianges, and the durability enabling it to resist decay. Pine 
and redwood resist decay very well, but are so soft that they are 
badly cut by the rail-fianges and do not hold the spikes very 
well, necessitating frequent respiking. Since the spikes must 
be driven within certain very limited areas on the face of each 
tie, it does not require many spike-holes to ''spike-kilP' the 
tie. On sharp curves, especially with heavy traffic, the wheel- 
flange pressure produces a side pressure on the rail tending to 
overturn it, which tendency is resisted by the spike, aided some- 
times by rail-braces. Whenever the pressure becomes too great 
the spike will yield somewhat and will be slightly withdraw^n. 
The resistance is then somewhat less and the spike is soon so 
loose that it must be redriven in a new^ hole. If this occurs 
very often, the tie may need to be replaced long before any decay 
has set in. When the traffic is very light, the wood very dura- 
ble, and the climate favorable, ties have been known to last 
25 years. 

205. Dimensions. The usual dimensions for the best roads 
(standard gauge) are 8' to 8' 6'' long, 6'' to 7" thick, and S" to 
10'' wide on top and bottom (if they are hewed) or 8'' to 9'' 
wide if they are sawed. For cheap roads and light traffic the 
length is shortened sometimes to 7' and the cross-section also 
reduced. On the other hand a very few roads use ties 9' long. 

Two objections are urged against saAved ties: r -^st, that the 
grain is torn by the saw, leaving a woolly surface w^hich induces 
decay; and secondly, that, since timber is not perfectly straight- 



§ 206. TIES. 223 

grained, some of the fibers are cut obliquely, exposing their ends, 
which are thus hable to decay. The use of a " planer-saw" ob- 
viates the first difficulty. Chemical treatment of ties obviates 
both of these difficulties. Sawed ties are more convenient to 
handle, are a necessity on bridges and trestles, and it is even 
claimed, although against commonly received opinion, that 
actual trial has demonstrated that they are more durable than 
hewed ties. 

206. Spacing. The spacing is usually 14 to 16 ties to a 30- 
foot rail. This number is sometimes reduced to 12 and even 
10, and on the other hand occasionally increased to 18 or 20 by 
employing narrower ties. There is no economy in reducing the 
number of ties very much, since for any required stiffness of 
track it is more economical to increase the number of supports 
than to increase the weight of the rail. The decreasing cost of 
rails and the increasing cost of ties have materially changed the 
relation between number of ties and weight of rail to produce a 
given stiffness at minimum cost, but many roads have found it 
economical to employ a large number of ties rather than increase 
the weight of the rail. On the other hand there is a practical 
limit to the number that may be employed, on account of the 
necessary space between the ties that is required for proper 
tamping. This width is ordinarily about twice the width of the 
tie. At this rate, w^th light ties 6'' wide and w^ith 12" clear 
space, there would be 20 ties per 30-foot rail, or 3520 per mile. 
The smaller ties can generally be bought much cheaper (propor- 
tionately) than the larger sizes, and hence the economy. 

Track instructions to foremen generally require that the 
spacing of ties shall not be uniform along the length of any 
rail. Since the joint is generally the weakest part of the rail 
structure, the joint requires more support than the center of the 
rail. Therefore the ties are placed with but 8'' or 10" clear 
space between them at the joints, this applying to 3 or 4 ties at 
each joint; the remaining ties, required for each rail length, are 
equally spaced along the remaining distance. 

207. Specifications. The specifications for ties are apt to 
include the items of size, kind of wood, and method of construc- 
tion, besides other minor directions about time of cutting, sea- 
soning, delivery, quality of timber, etc. 

(a) Size. The particular size or sizes required will be some^ 
what as indicated in § 205. 



224 RAILROAD CONSTRUCTION. § 208. 

(b) Kind of wood. When the kind or kinds of wood are 
specified, the most suitable kinds that are available in that 
section of country are usually required. 

(c) Method of construction. It is generally specified that tlie 
ties shall be hewed on two sides; that the two faces thus made 
shall be parallel planes and that the bark shall be removed. It 
is sometimes required that the ends shall be sawed off square; 
that the timber shall be cut in the winter (when the sap is down) ; 
and that the ties shall be seasoned for six months These last 
specifications are not required or lived up to as much as their 
importance deserves. It is sometimes required that the ties shall 
be delivered on the right of way, neatly piled in rows, the alter- 
nate rows at right angles, piled if possible on ground not lower 
than the rails and at least seven feet away from them, the lower 
row of ties resting on two ties w^hich are themselves supported 
so as to be clear of the ground. 

(d) Quality of timber. The usual specifications for sound 
timber are required, except that they are not so rigid as for a 
better class of timber work The ties must be sound, reason- 
ably straight-grained, and not very crooked — one test being that 
a line joining the center of one end with the center of the middle 
shall not pass outside of the other end. Splits or shakes, espe- 
cially if severe, should cause rejection. 

Specifications sometimes require that /he ties shall be cut 

from single trees, making 
M^^^ what is known as ^^pole 
■\iM§^k ties'' and definitely con- 




L_ __„_^^L (^emning- those which are 

POLE TIE. SLAB TIE. QUAqrER TIE. ,. „ 

^ ,, ^ cut or split from larger 

FiQ. 109. — Methods of cutting Ties. , , . . , /^ i i 

trunks, giving two slab 

ties" or four '' quarter ties'' for each cross- sect ion, as is illustrated 
in Fig. 109. Even if pole ties are better, their exclusive use 
means the rapid destruction of forests of young trees. 

2o8. Regulations for laying and renewing ties. The regula- 
tions issued by railroad companies to their track foremen will 
generally include the following, in addition to directions regard- 
ing dimensions, spacing, and specifications given in §§ 204-207. 
When hewn ties of somewhat variable size are used, as is fre- 
quently the case, the largest and best are to be selected for use 
as joint ties. If the upper surface of a tie is found to be warped 
(contrary to the usual specifications) so that one or both rails do 



§ 209. TIES. 225 

not get a full bearing across the whole width of the tie, it must 
be adzed to a true surface along its whole length and not merciv 
notched for a rail-seat. When respiking is necessary and spikes 
have been pulled out, the holes should be immediately plugged 
with ^'wooden spikes," which are supplied to the foreman for 
that express purpose, so as to fill up the holes and prevent the 
decay which would otherwise take place when the hole becomes 
filled with rain-water. Ties should always be laid at right angles 
to the rails and never obliquely Minute regulations to prevent 
premature rejection and renewal of ties are frequently made. It 
is generally required that the requisitions for renewals shall be 
made by the actual coimt of the individual ties to be rencAved 
instead of by any wholesale estimates. It is unwise to have ties 
of widely variable size, hardness, or durability adjacent to each 
other in the track, for the uniform elasticity, so necessar}- for 
smooth riding^ will be unobtainable under those circumstances. 

209. Cost of ties. ¥,Tien railroads can obtain ties cut by 
farmers from woodlands in the immediate neighborhood, the 
price will frequently be as low as 20 c for the smaller sizes^ 
running up to 50 c for the larger sizes and better qualities, espe- 
cially when the timber is not ver}^ plentiful Sometimes if a 
railroad cannot procure suitable ties from its immediate neigh- 
borhood, it will find that adjacent railroads control all adjacent 
sources of supply for their own use and that ties can only be 
procured from a considerable distance, Txdith a considerable added 
cost for transportation . First-class oak ties cost about 75 to 80 c. 
and frequently much more Hemlock ties can generally be 
obtained for 35 c. or less. 

PRESERVATIVE PROCESSES FOR WOODEN TIES. 

210. General principle. Wood has a fibrous cellular struc- 
ture, the cells being filled with sap or air. The woody fiber is 
but little subject to decay unless the sap undergoes fermentation. 
Preservative processes generally aim at removing as much of the 
water and sap as possible and filling up the pores of the wood 
with an antiseptic compound The most common methods (ex- 
cept one) all agree in this general process and only differ in the 
method employed to get rid of the sap and in the antiseptic 
chemical with which the fibers are filled One valuable feature 
of these processes lies in the fact that the softer cheaper woods 



220 RAILROAD CONSTRUCTION. §211. 

(such as hemlock and pine) are more readily treated than are the 
harder woods and yet will produce practically as good a tie as a 
treated hard-wood tie and a very much better tie than an un- 
treated hard-wood tie. The various processes will be briefly 
described, taking up first the process which is fundamentally 
different from the others, viz., Vulcanizing^ 

211. Vulcanizing. The process consists in heating the timber 
to a temperature of 300° to 500° F. iii a cyhnder, the air being 
under a pressure of 100 to 175 lbs. per square inch. By this 
process the albumen in the sap is coagulated, the water evapo- 
rated, and the pores are partially closed by the coagulation of 
the albumen. It is claimed that the heat sterilizes the wood and 
produces chemical changes in the wood which give it an antisep- 
tic character. It has been very extensively used on the elevated 
lines of New York City, and it is claimed to give perfect satis- 
faction. The treatment has cost that road 25 c. per tie. 

212. Creosoting. This porcess consists in impregnatmg the 
wood with wood-creosote or with dead oil of coal-tar. Wood- 
creosote is one of the products of the destructive distillation of 
w^ood — usually long-leaf pine. Dead oil of coal-tar is a prod- 
uct of the distillation of coal-tar at a temperature between 480° 
and 760° F. It would require about 35 to 50 pounds of creo^ 
sote to completely fill the pores of a cubic foot of Avood But 
it w^ould be impossible to force such an amount into the wood, 
nor is it necessary or desirable. About 10 pounds per cubic 
foot, or about 35 pounds per tie, is all that is necessary. For 
piling placed in salt water about 18 to 20 pounds per cubic foot 
is used, and the timber is then perfectly protected against the 
ravages of the teredo navalis. To do the work, long cylinders, i 
which m^ay be opened at the ends, are necessary. Usually the \ 
timbers are run in and out on iron carriages running on rails 
fastened to braces on the inside of the cylinder. When the load ; 
has been run in, the ends of the cylinder are fastened on. The I 
water and air in the pores of the wood are first drawn out b\^ i 
subjecting the wood alternately to steam-pressure and to the ■ 
action of a vacuum-pump. This is continued for several hours, i 
Then, after one of the vacuum periods, the cylinder is filled > 
with creosote oil at a temperature of about 170° F The pumps 
are kept at work until the pressure is about 80 to 100 pounds 
per square inch, and is maintained at this pressure from one to 1 
two hours according to the size of the timber. The oil is then i 



§ 213. TIES. 227 

withdrawn, the cylinders opened, the train pulled out and an- 
other load made up in 40 to 60 minutes. The average time re- 
quired for treating a load is about 18 or 20 hours, the absorption 
about 10 or 11 pounds of oil per cubic foot, and the cost (1894) 
from $12.50 to $14.50 per thousand feet B. M 

213. Burnettizing (chloride-of-zinc process). This process is 
very similar to the creosoting process except that the chemical is 
chloride of zinc, and that the chemical is not heated before use. 
The preliminary treatment of the wood to alternate vacuum and 
pressure is not continued for quite so long a period as in the 
creosoting process. Care must be taken, in using this process, 
that the ties are of as uniform quality as possible, for seasoned 
ties vdW absorb much more zinc chloride than unseasoned !j"n the 
same time), and the product will lack uniformity unless the sea- 
soning is uniform. The A., T. & S. Pe R. R. has works of its 
own at which ties are treated by this process at a cost of about 
25 c. per tie. The Southern Pacific R. R. also has works for 
burnettizing ties at a cost of 9.5 to 12 c per tie The zinc- 
chloride solution used in these works contains only 1.7% of zinc 
chloride instead of over 3% as used in the Santa Fe works, which 
perhaps accounts partially for the great difference in cost per tie. 
One great objection to burnettized ties is the fact that the chem- 
ical is somewhat easily washed out, when the wood again be- 
comes subject to decay Another objection, which is more 
forcible with respect to timber subject to great stresses^ as in 
trestles, than to ties, is the fact that when the solution of zinc 
chloride is made strong (over 3%) the timber is made very brittle 
and its strength is reduced. The reduction in strength has been 
shown by tests to amount to J to -^^ of the ultimate strength, 
and that the elastic limit has been reduced by about 4. 

214. Kyanizing (bichloride-of- mercury or corrosive-sublimate 
process). This is a process of "steeping.'' It requires a much 
longer time than the previously described processes, but does not 
require such an expensive plant. Wooden tanks of sufficient 
size for the timber are all that is necessary. The corrosive subli- 
mate is first made into a concentrated solution of one part of 
chemical to six parts of hot water. When used in the tanks this 
solution is weakened to 1 part in 100 or 150. The wood will 
absorb about 5 to 6.5 pounds of the bichloride per 100 cubic 
feet, or about one pound for each 4 to 6 ties. The timber is 
allowed to soak in the tanks for several days, the general rule 



228 RAILROAD CONSTRUCTION. § 215. 

being about one day for each inch of least thickness and one day- 
over — which means seven days for six-inch ties, or thirteen (to 
fifteen) days for 12'' timber (least dimension). The process is 
somewhat objectionable on account of the chemical being such a 
virulent poison, workmen sometimes being sickened by the fumes 
arising from the tanks. On the Baden railway (Germany) 
kyanized ties last 20 to 30 years. On this railwa}^ the wood is 
always air-dried for two weeks after impregnation and before 
being used, which is thought to have an important effect on its 
durability. The solubility of the chemical and the hability of 
the chemical washing out and lea^'ing the wood unprotected is 
an element of weakness in the method. 

215. Wellhouse (or zinc- tannin) process. The last Uvo 
methods described (as well as some others employing similar 
chemicals) are open to the objection that since the wood is im- 
pregnated with an aqueous solution, it is liable to be washed out 
very rapidly if the wood is placed under water, and will even 
disappear, although more slowly, under the action of moisture 
and rain. Several processes have been proposed or patented to 
prevent this. Man}^ of them belong to one class, of which the 
Wellhouse process is a sample. By these processes the timber 
is successively subjected to the action of two chemicals, each 
individually soluble in water, and hence readily impregnating 
the timber, but the chemicals when brought in contact form in- 
soluble compounds which cannot be washed out of the wood- 
cells. By the Wellhouse process, the wood is first impregnated 
with a solution of chloride of zinc and glue, and is then subjected 
to a bath of tannin under pressure. The glue and tannin com- 
bine to form an insoluble leathery compound in the cells, which 
will prevent the zinc chloride from being washed out. It is 
being used by the A., T. &. S. Fe R. R., their works being 
located at Las Vegas, New Mexico, and also by the Union 
Pacific R. R. at their works at Laramie, Wyo. In 1897 Mr. J. 
M. Meade, a resident engineer on the A., T. & S. Fe, exhibited 
to the Roadmasters Association of America a piece of a tie treated 
by this process which had been taken from the tracks after 
nearly 13 years' service. The tie was selected at random, was 
taken out for the sole purpose of having a specimen, and was 
still in sound condition and capable of serving many years longer. 
The cost of the treatment was then quoted as 13 c. per tie. 



§ 216. TIES. « 229 

It was claimed that the treatment trebled the life of the tie 
besides adding to its spike-holding power. 

216. Cost of treating. The cost of treating ties by the vari- 
ous methods has been estimated as follows * — assuming that 
the plant was of sufficient capacity to do the work economi- 
cally: creosoting, 25 c. per tie; vulcanizing, 25 c. per tie; 
burnettizing (chloride of zinc), 8.25 c. per tie; kyanizing (steep- 
ing in corrosive sublimate), 14.6 c. per tie; Wellhouse process 
(chloride of zinc and tannin), 11.25 c. per tie. These estimates 
are only for the net cost at the wqrks and do not include the 
cost of hauling the ties to and from the works, which may mean 
5 to 10 c. per tie. Some of these processes have been installed 
on cars which are transported over the road and operated where 
most convenient. 

217. Economics of treated ties. The fact that treated ties are 
not universall}^ adopted is due to the argument that the added 
life of the tie is not worth the extra cost. If ties can be bought 
for 25 c, and cost 25 c. for treatment, and the treatment only 
doubles their life, there is apparently but little gained except 
the work of placing the extra tie in the track, which is more 
or less offset by the interest on 25 c. for the life cf the untreated 
tie, and the larger initial outlay makes a stronger impression on 
the mind than the computed ultimate economy. But when 
ties cost 75 c. and treatment costs onl}^ 25 c, or perhaps less, 
then the economy is more apparent and unquestionable. But 
this analysis may be made more closely. As shown in § 202, 
the disturbance of the roadbed on account of frequent renewals 
of untreated ties is a disadvantage which would justif}' an appre- 
ciable expenditure to avoid, although it is ^^ery difficult to 
closely estimate its true value. The annual cost of a system of 
ties may be considered as the sum of (a) the interest on the first 
cost, (b) the annual sinking fund that would buy a new tie at 
the end of its life, and (c) the average annual cost of mainte- 
nance for the life of the tie, which includes the cost of laying and 
the considerable amount of subsequent tamping that must l)e 
done until the tie is fairly settled in the roadbed, besides the 
regular trackwork on the tie, which is practically constant. This 
last item is difficult to compute, but it is easy to see that, since 



* Bull. No. 9, U. S. Dept. of Agric, Div. of Forestry. App. No. 1, by 
Henry Flad. 



230 RAILROAD CONSTRUCTION. § 217. 

the cost of laying the tie and the subsequent tamping to obtain 
proper settlement is the same for all ties (of similar form), the 
average annual charge on the longer-lived tie would be much less. 
In the following comparison item (c) is disregarded, simply re- 
membering that the advantage is with the longer-lived tie. 



Untreated tie. 

Original cost 40 cents 

Life (assumed at) 7 years 

Item (a) — interest on first cost @, 4^ 1.6 cents 

" ih) — sinking fund (^ 4?o 5.1 " 

" (c) — (considered here as balanced) 



Average annual cost (except item (c)) 6.7 cent 



Treated tie, 
65 cents 
14 years 



2.6 cents 
3.6 *• 



6.2 cents 



'^ On this basis treated ties will cost 0.5 cent less per annum 
besides the advantage of item (c) and the still more indefinite 
advantages resulting from smoother running of trains, less wear ^ 
and tear on rolling stock, etc., due to less disturbance of the ' ^ 
roadbed. ' ^' 

In Europe, where wood is expensive, untreated ties are 
seldom used, as the treatment is alwajs considered to be worth 
more than it costs. The rapid destruction of the forests of tim- ' ^ 
ber in this country is having the effect of increasing the price, so ^'1 
that it will not belong before treated ties (or metal ties) will be 
economical for a large majority of the railroads of the country. 

{Note added in 1902.) Some modifications of the above 
processes have been devised in recent years, among them 
being the ' ^•^' 

n 

Creo-resinate process — creosote, resin, and formaldehyde; "' w' 

Water-creosote " — emulsion of creosote and water; } fre 

Zinc-creosote " — emulsion of creosote and zinc -chloride; 

AUardj^ce *' - — injection of chloride of zinc followed by creosote; 

Hasselmann *' — boiling in sulphates of iron, copper, etc. 



The Atchison, Topeka and Santa Fe R, R. has compiled a 
record of treated pine ties remo\'ed in 1897, '98, '99, and 1900,. 
showing that the average life of the ties removed had been about 
11 years. On the Chicago, Rock Island and Pacific R. R., the' 
average life of a very large number of treated hemlock and ' 
tamarack ties was found to be 10.57 years. Of one lot of 21850 ' 
ties, 12% still remained in the track after 15 years' exposure. " 

It has been demonstrated that much depends on the minor 



ine 



§ 218. TIES. 231 

details of the process — whatever it may be. As an illustra- 
tion, an examination of a batch of ties, treated by the zinc- 
creosote process, showed 84% in service after 13 years' ex- 
posure; another batch, treated by another contractor by the 
same process (nominally), showed 50% worthless after a service 
of six years. 

METAL TIES. 

2i8. Extent of use. In 1894 "^ there were nearly 35000 miles 
of ''metal track" in various parts of the world. Of this total, 
there were 3645 miles of ''longitudinals" (see § 224), found ex- 
clusively in Europe, nearly all of it being in Germany. There 
were over 12000 miles of "bowls and plates" (see § 223), found 
almost entirely in British India and in the Argentine Republic. 
The remainder, over 18000 miles, was laid with metal cross-ties 
of various designs. There were over 8000 miles of metal cross- 
ties in Germany alone, about 1500 miles in the rest of Europe, 
over 6000 miles in British India, nearly 1000 miles in the rest 
of Asia, and about 1500 miles more in various other parts of the 
world. Several railroads in this country have tried various de- 
signs of these ties, but their use has never passed the experi- 
mental stage. These 35000 miles represent about 9% of the 
total railroad mileage of the world — nearly 400000 miles. They 
represent about 17.6% of the total railroad mileage, exclusive of 
the United States and Canada, where they are not used at all, 
except experimentally. In the four years from 1890 to 1894 the 
use of metal track increased from less than 25000 miles to nearly 
I 35000 miles. This increase was practically equal to the total in- 
crease in railroad mileage during that time, exclusive of the 
increase in the United States and Canada. This indicatss a 
large growth in the percentage of metal track to total mileage, 
and therefore an increased appreciation of the advantages to be 
derived from their use. 

219. Durability. The durability of metal ties is still far 
from being a settled question, due largely to the fact that the 
best form for such ties is not yet determined, and that a large 
. part of the apparent failures in metal ties have been evidently 
due to defective design. Those in favor of them estimate the 
life as from 30 to 50 years. The opponents place it at not more 

* Bulletin No. 9, U. S. Dept. of Agriculture, Div. of Forestry. 



232 RAILROAD CONSTRUCTION. § 220. 

than 20 years, or perhaps as long as the best of wooden ties. 
UnUke the wooden tie, however, which deteriorates as much 
with time as with usage, the Ufe of a metal tie is more largely a 
function of the traffic. The life of a well-designed metal tie has 
been estimated at 150000 to 200000 trains; for 20 trains per 
day, or say 6000 per year, this would mean from 25 to 33 years. 
20 trains per day on a single track is a much larger number than 
will be found on the majority of railroads. Metal ties are found 
to be subject to rust, especially when in damp localities, such as 
tunnels; but on the other hand it is in such confined localities, 
where renewals are troublesome, that it is especially desirable to 
employ the best and longest-lived ties. Paint, tar, etc., have 
been tried as a protection against rust, but the efficacy of such 
protection is as yet uncertain, the conditions preventing any re- 
newal of the protection — such as may be done by repainting a 
bridge, for example. Failures in metal cross-ties have been 
largely due to cracks which begin at a corner of one of the square 
holes which are generally punched through the tie, the holes 
being made for the bolts by which the rails are fastened to the 
tie. The holes are generally punched because it is cheaper. 
Reaming the holes after punching is thought to be a safeguard 
against this frequent cause of failure. Another method is to 
round the corners of the square punch with a radius of about 
y\ If a crack has already started, the spread of the crack may 
be prevented by drilling a small hole at the end of it. 

220. Form and dimensions of metal cross- ties. Since stability 
in the ballast is an essential quality for a tie, this must be accom- 
plished either by turning down the end of the tie or by ha^'ing 
some form of lug extending downward from one or more points 
of the tie. The ties are sometimes depressed in the center (see 
Plate VI, N. Y. C. & H. R. R. R. tie) to allow for a thick cover- 
ing of ballast on top in order to increase its stability in the 
ballast. This form requires that the ties should be sufficiently 
well tamped to prevent a tendency to bend out straight, thus 
widening the gauge. Many designs of ties are objectionable 
because they cannot be placed in the track without disturbing 
adjacent ties. The failure of many metal cross-ties, otherwise 
of good design, may be ascribed to too light weight. Those 
weighing much less than 100 pounds have proved too light. 
From 100 to 130 pounds weight is being used satisfactorily on 
German railroads. The general outside dimensions are about 




(Til jnre iiiigc 2: 



^ 



§ 221. TIES. 233 

the same as for wooden ties, except as to thickness. The metal 
is generall}^ from Y^ to f '' thick. They are, of course, only made 
of wrought iron or steel, cast iron being used only for ^' bowls" or 
^^ plates'' (see § 223). The details of construction of some of the 
most commonly used ties may be seen by a study of Plate VI. 

221. Fastenings. The devices for fastening the rails to the 
ties should be such that the gauge may be widened if desired on 
curves, also that the gauge can be made true regardless of slight 
inaccuracies in the manufacture of the ties, and also that shims 
may be placed under the rail if necessary during cold weather 
when the tie is frozen into the ballast and cannot be easily 
disturbed. Some methods of fastening require that the base of 
the rail be placed against a lug which is riveted to the tie or 
wliich forms a part of it. This has the advantage of reducing 
the number of pieces, but is apt to have one or more of the 
disadvantages named above. Metal keys or wooden wedges are 
sometimes used, but the majority of designs employ some form 
of bolted clamp. The form adopted for the experimental ties 
used by the N. Y. C. & H. R. R. R. (see Plate VI) is especially 
ingenious in the method used to vary the gauge or allow for 
inaccuracies of manufacture. Plate VI shows some of the 
methods of fastening adopted on the principal types of ties. 

222. Cost. The cost of metal cross-ties in Germany averages 
about 1.6 c. per pound or about $1.60 for a 100-lb. tie. The ties 
manufactured for the N. Y. C. & H. R. R. R. in 1892 weighed 
about 100 lbs. and cost $2.50 per tie, but if they had been made 
in larger quantities and with the present price of steel the cost 
would possibly have been much lower. The item of freight 
from the place of manufacture to the place where used is no 
inconsiderable item of cost with some roads. Metal cross-ties 
have been used by some street railroads in this country. Those 
used on the Terre Haute Street Railway weigh 60 pounds and 
cost about 66 c. for the tie, or 74 c. per tie with the fastenings. 



223. Bowls or plates. As mentioned before, over 12000 miles 
of railway, chiefly in British India and in the Argentine Repub- 
lic, are laid with this form of track. It consists essentially of 
large cast-iron inverted ^' bowls" laid at intervals under each 
rail and opposite each other, the opposite bowls being tied 
together with tie-rods. A suitable chair is riveted or bolted on 
to the top of each bowl so as to properly hold the rail. Being 



234 RAILROAD CONSTRUCTION. § 224. 

made of cast iron, they are not so subject to corrosion as steel 
or wrought iron. They have the advantage that when old and 
worn out their scrap value is from 60% to 80% of their initial 
cost, while the scrap value of a steel or wrought-iron tie is prac- 
tically nothing. Failure generally occurs from breakage, the 
failures from this cause in India being about 0.4% per annum. 
They weigh about 250 lbs. apiece and are therefore quite expen- 
sive in first cost and transportation charges. There are miles 
of them in India which have already lasted 25 years and are 
still in a serviceable condition. Some illustrations of this form 
of tie are show^n in Plate VI. 

224. Longitudinals.* This form, the use of which is con- 
fined almost exclusively to Germany, is being gradually replaced 
on many lines by metal cross-ties. The system generally con- 
sists of a compound rail of several parts, the upper bearing rail 
being very light and supported throughout its length by other 
rails, which are suitably tied together with tie-rods so as to 
maintain the proper gauge, and which have a sufficiently broad 
base to be properly supported in the ballast. One great objec- 
tion to this method of construction is the 
difficulty of obtaining proper drainage espe- 
cially on grades, the drainage having a ten- 
dency to follow along the lines of the rails. 
;- ?yy.vyyyyyyy^ Thc coustruction Is mucli more complicated 
on sharp curves and at frogs and switches. 
Another fundamentally different form of 
longitudinal is the Haarman compound '' self -bearing '^ rail, 
having a base 12" wide and a height of 8'', the alternate sections 
breaking joints so as to form a practically continuous rail. 

Some of the other forms of longitudinals are illustrated in 
Plate VI. 

For a very complete discussion of the subject of metal ties, 
see the ''Report on the Substitution of Metal for Wood in 
Railroad Ties*' by E. E. Russell Tratman, it being Bulletin. 
No. 4, Forestry Division of the U. S. Dept. of Agriculture. 

* Although the discussion of longitudinals might be considered to be 
long more propeily to the subject of RATLs.yet the essential idea of ail de- 
signs must necessarily be the support 01 a rail-head on which the rolling 
stock may run, and therefore this form, unused in this country, will be 
briefly described hf re. 



CHAPTER IX. 

KAILS. 

225. Early forms. The first rails ever laid were wooden 
stringers which ^^ere used on very short tram-roads around coal- 
mines. As the necessity for a more durable rail increased, 
owing chiefly to the invention of the locomotive as a motive 
power, there were invented successively the cast-iron "fish- 
belly" rail and various forms of wrought-iron strap rails which 
finally developed into the T rail used in this country and the 
double-headed rail, supported by chairs, used so extensively in 
England. The cast-iron rails were cast in lengths of about 3 
feet and were supported in iron chairs wliich Avere sometimes 
set upon stone piers. A great deal of the first railroad track 
of this country was laid with longitudinal stringers of wood 
placed upon cross-ties, the inner edge of the stringers being 
protected by wrought-iron straps. The "bridge" rails were 
first roiled in this country in 1844. The "pear" section was 
an approach to the present form, but was very defective on 
account of the difficulty of designing a good form of joint. The 
"Stevens" section was designed in 1830 by Col. Robert L. 
Stevens, Chief Engineer of the Camden and Amboy Railroad; 
although quite defective in its proportions, according to the 
present knowledge of the requirements, it is essentially the pres- 
ent form. In 1836, Charles Vignoles invented essentially the 
same form in England; this form is therefore known throughout 
England and Europe as the Yignoles rail. 

226. Present standard forms. The larger part of modern 
railroad track is laid with rails uhich arc either ^'T" rails or 
the double-headed or "bull-headed" rails which are carried in 
chairs. The double-headed rail was designed w^ith a symmetri- 
cal form with the idea that after one head had been worn out 
by traffic the rail could be reversed, and that its life would be 
practically doubled. Experience has sho\ATi that the \\ ear of the 

235 



236 



RAILROAD CONSTRUCTION. 



§ 226. 



rail in the chairs is very great; so much so that when one head 
has been worn out by traffic the whole rail is generally useless. 




^^ 




BALT. & OHIO R. R. 
QUJNCYR.R. 1843. "BULL-HEAD. 



1828. 



^P 




VIGNOLES. 1836. 



CAMDEN & AM BOY. STEPHENSON. "PEAR.»> 

1832. 1838. 





FISH-BELLY"— CAST IRON. 



w. 



^ 1 



CAST IRON. 




Fig. 111.- 



reynolds— 1767. 
-Early Forms of Rails. 



If the rail is turned over, the worn places, caused by the chairs, 
make a rough track and the rail appears to be more brittle and 
subject to fracture, possibly due to the crystallization that may 
have occurred during the previous usage and to the reversal of 
stresses in the fibers. Whatever the explanation, experience has 
demonstrated the ]act. The ''bull-headed" 
rail has the lower head onl}^ large enough to 
properly hold the wooden ke3's with which 
the rail is secured to the chairs (see Fig. 112) 
and furnish the necessary strength. The use 
of these rails requires the use of two cast- 
iron chairs for each tie. It is claimed that 
such track is better for heavy and fast traffic, but it is more 




Fig. 112. — Bull- 
headed Rail a-nd 
Chair. 



§ 226. 



RAILS. 



237 



expensive to build and maintain. It is the standard form of 
track in England and some parts of Europe. 

Until a fevv' years ago there Avas a very great multiplicity 
in the designs of "T'' rails as used in this countr^'^ nearly ever}^ 
prominent railroad having its own special design, which perhaps 
differed from that of some other road by only a very minute and 
insignificant detail, but vrhich nevertheless would require a 
complete new set of rolls for rolling. This certainly must have 
had a very appreciable effect on the cost of rails. In 1893, the 
American Society of Civil Engineers, after a ver}^ exhaustive 
investigation of the subject, extending over several years, hav- 
ing obtained the opinions of the best experts of the country, 
adopted a series of sections which have been verj- extensively 
adopted by the railroads of this country. Instead of having 
the rail sections for various weights to be geometrically similar 
figures, certain dimensions are made constant, regardless of the 
weight. It was decided that the metal should be distributed 
through the section in the proportions of — head 42%, web 21%, 
and flange 37%. The top of the head should have a radius of 




Fig. 113. — Am. Soc. C. E. Standard Rail Section. 



12"; the top corner radius of head should be y^; the lower 
corner radius of head should be ^g''; the corners of the flanges, 
y-.y'' radius; side radius of web, 12"; top and bottom radii of 
web corners, J"; and angles with the horizontal of the under side 



238 



RAILROAD CONSTRUCTION. 



§227. 



of the head and the top of the flange, 13°. The sides of the head 
are vertical. 

The height of the rail (D) and the width of the base (C) are 
always made equal to each other. 













Weight per Yarc^ 


I. 












40 


45 


50 


55 


60 


65 


70 


75 

2^r 


80 


85 


90 


95 


ICO 


A 


IF 


2" 


2r 


2¥' 


21- 


2M" 


2xV' 


2-^-'' 


2A" 


2F 


211" 


2r 


B 


11 


27 
^■4 


t\ 


a 


U 


h 


U 


•H 


U 


r^B 


1% 


t"^ 


T% 


C &D 


3+ 


3B 


H 


4l^H 


41 


4/e 


4f 


4i.^ 


5 


5,^ 


51 


5t§ 


5t 


E 


1 


Ih 


H 


i?. 


II 


If 


il 


§1 


1 


il 


II 


If 


M 


F 


IM 


m 


2i^ 


2U 


2U 


21 


2y 


2M 


2f 


2f 


211 


211 


s^\ 


G 


Uz 


ItV 


H 


Hi 


1^ 


li^ 


ly 


HI 


u 


HI 


HI 


111 


HI 



The chief features of disagreement among railroad men relate 
to the radius of the upper corner of the head a«nd the slope of the 
side of the head. The radius (t^^'O adopted for the upper corner 
(constant for all weights) is a little more than is advocated by 
those in favor of ^' sharp corners" who often use a radius of |". 
On the other hand it is much less than is advocated by those 
who consider that it should be nearly equal 
to (or even greater than) the larger radius 
universally adopted for the corner of the 
wheel-flange. The discussion turns on the 
relative rapidity of rail wear and the wear 
of the wheel-flanges as affected by the rela- 
tion of the form of the wheel-tread to that 
of the rail. It is argued that sharp rail 
corners wear the wheel-flanges so as to 
produce sharp flanges, which are liable to 
Fig. 114.— Relation cause derailment at switches and also to 
?read!'^^''^''''^'^ require that the tires of engine-drivers 
must be more frequently turned down to their true form. On 
the other hand it is generally beheved that rail wear is much less 
rapid while the area of contact between the rail and wheel-flange 
is small, and that when the rail has worn down, as it invariably 
does, to nearly the same form as the wheel-flange, the rail ^\ ears 
away very quickly. 

227. Weight for various kinds of traffic. The heaviest rails 
in regular use weigh 100 lbs. per yard, and even these are only 
used on some of the heaviest traffic sections of such roads as the 




§ 228. RAILS. 239 

N. Y. Central, the Pennsylvania, the N. Y., N. H. & H., and 
a few others. Probabh^ the larger part of the mileage of the 
country is laid with 60- to 75-lb. rails — considering the fact that 
''the larger part of the mileage" consists of comparatively light- 
traffic roads and may exclude all the heavy trunk lines. Very 
light-traffic roads are sometimes laid with 56-lb. rails. Roads 
with fairly heavy traffic generally use 75- to 85-lb. rails, espe- 
cially when grades are heavy and there is much and sharp curva- 
ture. The tendency on all roads is toward an increase in the 
\Aeight, rendered necessary on account of the increase in the 
weight and capacity of rolling stock, and due also to the fact that 
the price of rails has been so reduced that it is both better and 
cheaper to obtain a more solid and durable track by increasing 
the weight of the rail rather than by attempting to support a 
weak rail by an excessive number of ties or by excessive track 
labor in tamping. It should be remembered that in buying rails 
the mere weight is, in one sense, of no importance. The im- 
portant thing to consider is the strength and the stiffness. If 
we assume that all weights of rails hj.ve sirnilar cross-section, 
(which is nearly although not exactl}^ true), then, since for beams 
of similar cross-sections the strength varies as the a.ihe of the 
homologous dimensions and the stiffness as the fourth poivers 
while the area (and therefore the weight per unit of length) 
only varies as the square, it follows that the stiffness varies as 
the square of the \^ eight, and the strength as the j power of the 
weight. Since for ordinary variations of weight the price per 
ton is the same, adding (say) 10% to the weight (and cost) adds 
21% to the stiffness and over 15% to the strength. As another 
illustration, using an 80-lb. rail instead of a 75-lb. rail adds only 
6f % to the cost, but adds about 14% to the stiffness and nearly 
11% to the strength. This shows why heavier rails are more 
economical and are being adopted even when they are not abso- 
lutel}^ needed on account of heavier rolling stock. The stiffness, 
strength, and consequent durability are increased in a much 
greater ratio than the cost. 

228. Effect of stiffness on traction. A very important but 
generally unconsidered feature of a stiff rail is its effect on trac- 
tive force. An extreme illustration of this principle is seen 
when a vehicle is drawn over a soft sandy road. The constant 
compression of the sand in front of the wheel has virtually the 
same effect on traction as drawing the wheel up a grade whose 



240 IUILi:OAD CONSTRUCTION. § 229. 

steepness depends on the radius of the wheel and the depth of 
the rut. On the other hand, if a wheel, made of perfectly 
elastic material, is rolled over a surface which, while supported 
with absolute rigidity, is also perfectly elastic, there would be a 
forward component, caused by the expanding of the compressed 
metal just behind the center of contact, which would just bal- 
ance the backward component. If the rail w^as supported 
throughout its length by an absolutely rigid support, the high 
elasticity of the wheel-tires and rails would reduce this form of 
resistance to an insignificant quantity, but the ballast and even 
the ties are comparatively inelastic. When a weak rail yields, 
the ballast is more or less compressed or displaced, and even 
though the elasticit}^ of the rail brings it back to nearly its 
former place, the work done in compressing an inelastic material 
is wholly lost. The effect of this on the fuel account is certainly 
very considerable and yet is frequently entirely overlooked. It 
is practically impossible to compute the saving in tractive power, 
and therefore in cost of fuel, resulting from a given increase in 
the weight and stiffness of the rail, since the yielding of the rail 
is so dependent on the spacing of the ties, the tamping, etc. But 
it is not difficult to perceive in a general way that such an econ- 
omy is possible and that it should not be neglected in considering 
the value of stiffness in rails. 

229. Length of rails. The standard length of rails with most 
railroads is 30 feet. In recent years many roads have been try- 
ing 45-foot and even 60-foot rails. The argument in favor of 
longer rails is chiefly that of the reduction in track-joints, which 
are costl}^ to construct and to maintain and are a fruitful source 
of accidents. Mr. Morrison of the Lehigh Valley R. R.* declares 
that, as a result of extensive experience with 45-foot rails on 
that road, he finds that they are much less expensive to handle, 
and that, being so long, they can be laid around sharp curves 
without being curved in a machine, as is necessary with the 
shorter rails. The great objection to longer rails lies in the 
difficulty in allowing for the expansion, which will require, in 
the coldest weather, an opening at the joint of nearly f for a 
60-foot rail. The Pennsylvania R. R. and the Norfolk and 
Western R. R. each have a considerable mileage laid with 60-foot 
rails. 

* Report, Roadmasters Association, 1895. 



§ 230. RAILS. 241 

230. Expansion of rails. Steel expands at the rate of .0000065 
of its length per degree Fahrenheit. The extreme range of tem- 
perature to which any rail will be subjected wiU be about 160°, 
or sa}^ from -20° F. to +140° F. With the above coefficient 
and a rail length of 60 feet the expansion would be 0.0624 foot, 
or about | inch. But it is doubtful whether there would ever 
be such a range of motion even if there were such a range of 
temperature. Mr. A. Torrey, cliief engineer of the Mich. Cent. 
R. R., experimented with a section over 500 feet long, which, 
although not a single rail, was made ^'continuous" by rigid 
splicing, and he found that there was no appreciable additional 
contraction of the rail at any temperature below +20° F. The 
reason is not clear, but the fact is undeniable. 

The heavy girder rails, used by the street railroads of the 
country, are bonded together with perfectly tight rigid joints 
which do not permit expansion. If the rails are laid at a tem- 
perature of 60° F. and the temperature sinks to 0°, the rails 
have a tendency to contract .00039 of their length. If this 
tendency is resisted by the friction of the pavement in wliich the 
rails are buried, it only results in a tension amounting to .00039 
of the modulus of elasticity, or say 10920 pounds per square 
inch, assuming 28 000 000 as the modulus of elasticity. This 
stress is not dangerous and may be permitted. If the tempera- 
ture rises to 120° F., a tendency to expansion and buckling will 
take place, wliich ^411 be resisted as before by the pavement, 
and a compression of 10920 pounds per square inch will be in- 
duced, which will likeAvise be harmless. The range of tempera- 
ture of rails which are buried in pavement is much less than 
when they are entirely above the ground and will probably 
never reach the above extremes. Rails supported on ties which 
are only held in place by ballast must be allowed to expand and 
contract almost freely, as the ballast cannot be depended on to 
resist the distortion induced by any considerable range of tem- 
perature, especially on curves. 

231. Rules for allowing for temperature. Track regulations 
generally require that the track foremen shall use iron {not 
wooden) shims for placing between the ends of the rails while 
splicing them. The thickness of these shims should A^ary with 
the temperature. Some roads use such approximate rules as the 
following : ^' The proper thickness for coldest weather is f^ of an 
inch; during spring and fall use ^ of an inch, and in the very 



242 



RAILROAD CONSTRUCTIOX 



§ 232. 



hottest weather ^\ of an inch should be allowed.'^ This is on 
the basis of a 30-foot rail. When a more accurate adjustment 
than this is desired, it may be done by assuming some very high 
temperature (120° to 150° F.) as a maximum, when the joints 
should be tight; then compute in tabular form the spacing for 
each temperature, varying by 20°, allowing 0''.0468 (almost 
exactly //') for each 20° change. Such a tabular form would 
be about as follows (rail length 30 feet) : 



Temperature .... 


150^ 


130° 


110° 


90^ 


70° 


50° 


30° 


10° 


-10° 


-30° 


Rail opening .... 





3 // 

64 


^Y 


i^'' 


3 '/ 
16 


ir 


^%'' 


21V 


r 


2 7// 
54 



One practical difficulty in the way of great refinement in this 
work is the determination of the real temperature of the rail 
when it is laid. A rail lying in the hot sun has a very much 
higher temperature than the air. The temperature of the rail 
cannot be obtained even by exposing a thermometer directly to 
the sun, although such a result might be the best that is easily 
obtainable. On a cloud}^ or rainy day the rail has practically 
the same temperature as the air; therefore on such days there 
need be no such trouble. 

232. Chemical composition. About 98 to 99.5% of the com- 
position of steel rails is iron, but the value of the rail, as a rail, 
is almost wholly dependent upon the large number of other 
chemical elements which are, or may be, present in very small 
amounts. The manager of a steel-rail mill once declared that 
their aim was to produce rails having in them — 

Carbon 0.32 to 0.40% 

Silicon 0.04 to 0.06% 

Phosphorus 0.09 to 0. 105% 

Manganese 1 .00 to 1 .50% 



The analysis of 32 specimens of rails on the Chic, Mil. & St. 
Paul R. R. showed variations as follows: 

Carbon 0.211 to 0.52% 

Silicon. 0.013 to 0.256% 

Phosphorus 0.055 to 0. 181% 

Manganese... 0.35 to 1.63%, 



§ 233 RAILS. 243 

These quantities have the same general relative proportions 
as the rail-mill standard given above, the differences lying 
mainly in the broadening of the limits. Increasing the per- 
centage of carbon by even a few hundredths of one per cent 
makes the rail harder, but likewise more brittle. If a track is 
well ballasted and not subject to heaving by frost, a harder and 
more brittle rail may be used without excessive danger of break- 
age, and such a rail will wear much longer than a softer tougher 
rail, although the softer tougher rail may be the better rail for 
a road having a less perfect roadbed. 

A small but objectionable percentage of sulphur is some- 
times found in rails, and very delicate analysis will often show 
the presence, in very minute quantities, of several other chem- 
ical elements. The use of a very small quantity of nickel or 
aluminum has often been suggested as a means of producing 
a more durable rail. The added cost and the uncertainty of 
the amount of advantage to be gained has hitherto prevented 
the practical use or manufacture of such rails. 

233. Testing. Chemical and mechanical testing are both 
necessary for a thorough determination of the value of a rail. 
The chemical testing has for its main object the determination 
of those minute quantities of chemical elements which have such 
a marked influence on the rail for good or bad. The mechanical 
testing consists of the usual tests for elastic limit, ultimate 
strength, and elongation at rupture, determined from pieces cut 
out of the rail, besides a ^'drop test." The drop test consists 
in dropping a weight of 2000 lbs. from a height of 16 to 20 feet 
on to the center of a rail which is supported on abutments, 
placed three or four feet apart. The number of blows required 
to produce rupture or to produce a permanent set of specified 
magnitude gives a measure of the strength and toughness of 
the rail. 

233a. Proposed standard specifications for steel rails. The 
following specifications for steel rails are those proposed by a 
committee of the American Railway Engineering and Main- 
tenance of Way Association in March, 1902: 

1. (a) Steel may be made b}^ the Bessemer or open-hearth 
process. 

(6) The entire process of manufacture and testing shall be in 
accordance Avith the best standard current practice, and special 
care shall be taken to conform to the following instructions: 



244 



RAILROAD CONSTRUCTION, 



§ 233. 



(c) Ingots shall be kept in a vertical position in pit-heating 
furnaces. 

(d) No bled ingots shall be used. 

(e) Sufficient material shall be discarded from the top of the 
ingots to insure sound rails. 



CHEMICAL PROPERTIES. 



2. Rails of the various weights per yard specified below shall 
conform to the following limits in chemical composition: 



Carbon 

Phosphorus shall not 
exceed 

Silicon shall not ex- 
ceed 

Manganese 



50 to 59 + 

lbs. 

per cent. 



0.35^.4>5 

0.10 

0.20 
0.70-1.00 



60 to 69 + 

lbs. 
per cent. 



0.38-0.48 

0.10 

0.20 
0.70-1.00 



70 to 79 + 

lbs. 
per cent. 



0.40-0.50 

0.10 

0.20 
0.75-1.05 



80 to 89 + 

lbs. 
per cent. 

0.43-0.53 

0.10 

0.20 
0.80-1.10 



90 to 100 

lbs. 
per cent. 



0.45-0.55 

0.10 

0.20 
0.80-1.10 



PHYSICAL PROPERTIES. 

3. One drop test shall be made on a piece of rail not more than 
6 feet long, selected from every fifth blow of steel. The test- 
piece shall be taken from the top of the ingot. The rail shall 
be placed head upwards on the supports and the various sections 
shall be subjected to the following impact tests: 



Weight of Rail in Pounds per Yard, 



45 to and including 55, 

More than 55 " " " 65, 

" 65 " " " 75, 

" 75 " " " 85. 

" 85 " *' " lOO 



Height of Drop 
in Feet. 




If any rail break when subjected to the drop test two additional 
tests will be made of other rails from the same blow of steel, and 
if either of these latter tests fail, all the rails of the blow which 
they represent will be rejected; but if both of these additional 
test-pieces meet the requirements all the rails of the blow which 
they represent will be accepted. If the rails from the tested 
blow shall be rejected for failure to meet the requirements of 



§ 233. RAILS. 245 

the drop test, as above specified, two other rails will be subjected 
to the same tests, oae from the blow next preceding and one from 
the blow next succeeding, the rejected blow. In case the first 
test taken from the preceding or succeeding blow shall fail two 
additional tests shall be taken from the same blow of steel, the 
acceptance or rejection of which shall also be determined as 
specified above, and if the rails of the preceding or succeeding, 
blow shall be rejected, similar tests may be taken from the pre- 
vious or following blows, as the case may be, until the entire 
group of five blows is tested, if necessary. The acceptance or 
rejection of all rails from any blow vnll depend upon the results 
of the tests thereof. 

HEAT TREATMENT. 

Th€ number of passes and speed of train shall be so regulated 
that on leaving the rolls at the final pass the temperature of the 
rail will not exceed that which requires a shrinkage allowance at 
the hot saws of 6 inches for 85-lb. and 6 J inches for 100-lb. rails, 
and no artificial means of cooling the rails shall be used between 
the finishing pass and the hot saws. 

TEST-PIECES AND METHODS OF TESTING. 

4. The drop-test machine shall have a tup of 2000 lbs. weight, 
the striking face of which shall have a. radius of not more than 
5 inches, and the test rail shall be placed head upwards on solid 
supports 3 feet apart. The anvil-block shall weigh at least 
20000 lbs., and the support shall be a part of, or firmly secured 
to, the anvil. 

5. The manufacturer shall furnish the inspector, daily, with 
carbon determinations of each blow, and a complete chemical 
analysis every 24 hours, representing the average of the other 
elements contained in the steel. These analyses shall be made 
on drillings taken from a small test ingot. 

FINISH. 

6. Unless otherwise specified the section of rail shall be the 
American standard, recommended by the American Society of 
Ci^^l Engineers, and shall conform, as accurately as possible, 
to the templet furnished by the railroad company, consistent 
with paragraph No. 7, relative to the specified weight. A vari- 



246 RAILROAD CONSTRUCTION. § 233. 

ation in height of /^^ inch less and 3^^ inch greater than the specified 
height will be permitted. A perfect fit of the splice-bars, how- 
ever, shall be maintained at all times. 

7. The weight of the rails shall be maintained as nearly as 
possible, after complying with paragraph No. 6, to that specified 
in contract. A variation of one-half of one per cent for an entire 
order will be allowed. Rails shall be accepted and paid for ac- 
cording to actual weights. 

8. The standard length of rails shall be 33 feet. Ten per cent 
of the entire order will be accepted in shorter lengths, var^ang 
by even feet down to 27 feet. A variation of ^ inch in length 
from that specified will be allowed. 

9. Circular holes for splice-bars shall be drilled in accordance 
with the specifications of the purchaser. The holes shall accu- 
rately conform to the drawing and dimensions furnished in every 
respect, and must be free from burrs. 

10. Rails shall be straightened while cold, smooth on head,, 
sawed square at ends, and, prior to shipment, shall have the 
burr, occasioned b}^ the saw-cutting, removed, and the ends 
made clean. No. 1 rails shall be free from injurious defects and 
flaws of all kinds. 

BRANDING. 

11. The name of the maker, the month and year of manu- 
facture shall be rolled in raised letters on the side of the web, 
and the number of the blow shall be stamped on each rail 



INSPECTION. 



i 



12. The inspector representing the purchaser shall have all 
reasonable facilities afforded to him by the manufacturer to 
satisfy him that the finished material is furnished in accord- 
ance with these specifications. All tests and inspections shall 
be made at the place of manufacture, prior to shipment. 

NO. 2 RAILS. 

13. Rails that possess any injurious physical defects, or which 
for any other cause are not suitable for first quality, or No. 1 
rails, shall be considered as No. 2 rails, provided, however, that 
rails which contain any physical defects which seriously impair 
their strength shall be rejected. The ends of all No. 2 rails 
shall be painted in order to distinguish them. 



§234. 



RAILS, 



247 




Fig. 115. 



234. Rail wear on tangents. When the wheel loads on a rail 
are abnormally heavy, and particularly when the rail has but 
little carbon and is unusually soft, the concentrated pressure 
on the rail is frequently greater than the 

elastic limit, and the metal ^^ flows" so that 
the head, although greatly abraded, will 
spread somewhat outside of its original lines, 
as shown in Fig. 115. The rail wear that 
occurs on tangents is almost exclusively 
on top. Statistics show that the rate of 
rail wear on tangents decreases as the rails 
are more worn. Tests of a large number of 
rails on tangents have shown a rail wear averaging nearly one 
pound per yard per 10 000 000 tons of traffic. There is about 
33 pounds of metal in one j^ard of the head of an 804b. rail. As 
an extreme value this may be worn down one-half, thus giving 
a tonnage of 165 000 000 tons for the life of the rail. Other 
estimates bring the tonnage doT\Ti to 125 000 000 tons. Since 
the locomotive is considered to be responsible for one-half (and 
possibly more) of the damage done to the rail, it is found that 
the rate of wear on roads with shorter trains is more rapid in 
proportion to the tonnage, and it is therefore thought that the 
life of a rail should be expressed in terms of the number of trains. 
This has been estimated at 300 000 to 500 000 trains. 

235. Rail wear on curves. On curves the maximum rail wear 
occurs on the inner side of the head of the outer rail, giving a 
worn form somewhat as sho^vn in Fig. 116. The dotted line 

shows the nature and progress of the rail wear 
on the inner rail of a curve. Since the press- 
ure on the outer rail is somewhat lateral 
rather than vertical, the ^'flow" does not 
take place to the same extent, if at all, on 
the outside, and whatever flow would take 
place on the inside is immediately worn off 
by the wheel-flange. Unlike the wear on 
tangents, the wear on curves is at a greater 
rate as the rail becomes more worn. 

The inside rail on curves wears chiefly on top, the same as 
on a tangent, except that the wear is much greater owing to the 
longitudinal slipping of the wheels on the rail, and the lateral 
slipping that must occur when a rigid four-wheeled truck is 




Fig. 116. 



248 KAILROAD CONSTRUCTION. § 236. 

guided around a curve. The outside rail is subjected to a 
greater or less proportion of the longitudinal slipping, likewise 
to the lateral slipping^ and, Avorst of all, to the grinding action 
of the flange of the wheel, which grinds off the side of the head 

The results of some very elaborate tests, made by Mr. A M. 
Wellington, on the Atlantic and Great Western R. R., on the 
wear of rails, seem to show that the rail wear on curves may be 
expressed by the formula: '"Total wear of rails on a d degree 
curve in pounds per yard per 10 000 000 tons duty = 1 +0 OSd^.'' 
'^It is not pretended that this formula is strictly correct even 
in theor}, but several theoretical considerations indicate that 
it may be nearly so." According to this formula the average 
rail wear on a 6° curve will be about twice the rail wear on a tan- 
gent. While this is approximately true^ the various causes 
modifying the rate of rail wear (length of trains, age and quality 
of rails, etc.) will result in numerous and large variations from 
the above formula, which should only be taken as indicating an 
approximate law. 

236. Cost of rails. In 1873 the cost of steel rails w^as about 
$120 per ton, and the cost of iron rails about $70 per ton 
Although the steel rails were at once recognized as superior to 
iron rails on account of more uniform wear, they w^ere an expen- 
sive luxury. The manufacture of steel rails by the Bessemer 
process created a revolution in prices, and they have steadily 
dropped in price until, during the last few years, steel rails have 
been manufactured and sold for $22 per ton. At such prices 
there is no longer any demand for iron rails, since the cost of 
manufacturing them is substantially the same as that of steel 
rails, while their durability is unquestionably inferior to that of 
steel rails. 



CHAPTER X. 
RAIL-FASTENINGS. 

RAIL-JOINTS 

237. Theoretical requirements for a perfect joint, A perfect 
rail-joint is one that has the same strength and stiffness — no 
more and no less — as the rails which it joins, and which w-ill 
not interfere with the regular and uniform spacing of ties. It 
should also be reasonably cheap both in first cost and in cost of 
maintenance. Since the action of heavy loads on an elastic rail 
is to cause a wave of translation in front of each wheel, any 
change in the stiffness or elasticity of the rail structure w^ill 
cause more or less of a shock, which must be taken up and 
resisted by the joint. The greater the change in stiffness the 
greater the shock, and the greater the destructive action of the 
shock. The perfect rail-joint must keep both rail-ends truly in 
line both laterally and vertically, so that. the flange or tread of 
the wheel need not jump or change its direction of motion sud~ 
denly in passing from one rail to the other. A consideration of 
all the above requirements will show^ that only a perfect wielding 
of rail-ends would produce a joint of uniform strength and stiff- 
ness which would give a uniform elastic w^ave ahead of each 
wheel. As welding is impracticable for ordinary railroad w^ork 
(see § 230), some other contrivance is necessary w^hich will 
approach this ideal as closely as may be. 

238. Efficiency of the ordinary angle-bar. Throughout the 
middle portion of a rail the rail acts as a continuous girder. If 
we consider for simplicity that the ties are unyielding, the deflec- 
tion of such a continuous girder between the ties will be but 
one-fourth of the deflection that would be found if the rail were 
cut half-way between the ties and an equal concentrated load 
were divided equally between the two unconnected ends. The 
maximum stress for the continuous girder would be but one-half 
of that in the cantilevers. Joining these ends wdth rail-joints 
will give the ordinary ^'suspended" joint. In order to main- 

249 



250 RAILROAD CONSTRUCTION. § 239. 

tain uniform strength and stiffness the angle-bars must supply 
the deficiency. These theoretical relations are modified to an 
unknown extent by the unknown and variable yielding of the 
ties From some experiments made by the Association of 
Engineers of Maintenance of Way of the P. R. R.* the following 
deductions were made* 

1. The capacity of a ^'suspended" joint is greater than that 
of a "supported'' joint — w^hether supported on one or three 
ties. (See § 240 ) 

2. That (with the particular patterns tested) the angle-bars 
alone can carry only 53 to 56% of a concentrated load placed 
on a joint. 

3. That the capacity of the whole joint (angle-bars and rail) 
is only 52 4% of the strength of the unbroken rail. 

4. That the ineffectiveness of the angle-bar is due chiefly to 
a deficiency in compressive resistance. 

Although it has been universally recognized that the angle- 
bar is not a perfect form of joint, its simplicity, cheapness, and 
reliability have caused its almost universal adoption. Within a 
very few years other forms (to be described later) have been 
adopted on trial sections and have been more and more extended, 
until their present use is very large. The present time (1900) is 
evidently a transition period, and it is quite probable that within 
a very few years the now common angle -plate will be as un- 
known in standard practice as the old-fashioned "fish-plate" 
is at the present time. 

239. Effect of rail gap at joints. It has been found that the 
jar at a joint is due almost entirely to the deflection of the joint 
and scarcely at all to the small gap required for expansion. 
This gap causes a drop equal to the versed sine of the arc having 
a chord equal to the gap and a radius equal to the radius of 
the wheel. Taking the extreme case (for a 30-foot rail) of a f 
gap and a 33'' freight-car wheel, the drop is about xoVo"- ^^ 
order to test how much the jarring at a joint is due to a gap be- 
tween the rails, the experiment was tried of cutting shallow 
notches in the top of an otherwise solid rail and running a loco- 
motive and an inspection car over them. The resulting jarring 
was practically imperceptible and not comparable to the jar pro- 
duced at joints. Notwithstanding this fact, many plans have 

* Roadniasters Association of America — Reports for 1897. 



§ 240. RAIL-FASTENIKGS. 261 

been tried for avoiding this gap. The most of these plans con- 
sist essentially of some form of compound rail, the sections 
breaking joints. (Of com'se the design of the compound rail 
has also several other objects in view.) In Fig. 117 are shown a 





Fig. 117. — Comt>ound Rail Sections. 

few of the very many designs which have been proposed. These 
designs have invariably been abandoned after trial. Another 
plan, which has been extensively tried on the Lehigh Valley 
R. R., is the use of mitered joints. The advantages gained by 
their use are as yet doubtful, while the added expense is unques- 
tionable. The '^ Roadmasters Association of America" in 1895 
adopted a resolution recommending mitered joints for double" 
track, but their use does not seem to be growing. 

240. " Supported," " suspended," and " bridge " joints. In a 
supported joint the ends of the rkils are on a tie. If the angle- 
plates are short, the joint is entireh^ supported on one tie; if 
very long, it may be possible to place three ties under one angle- 
bar and thus the joint is virtually supported on three ties rather 
than one. In a suspended joint the ends of the rails are midway 
between two ties and the joint is supported by the two. There 
ha^e always been advocates of both methods, but suspended 
joints are more generally used than supported joints. The 
opponents of three-tie joints claim that either the middle tie wiU 
be too strongly tamped, thus making it a supported joint, or 
that, if the middle tie is weakest, the joint becomes a very long 
(and therefore weak) suspended joint between the outer joint- 
ties, or that possibly one of the outer joint-ties gives way, thus 
breaking the angle-plate at the joint. Another objection which 
is urged is that unless the bars are very long (say 44 inches, as 
used on the Mich. Cent. R. R.) the ties are too close for proper 
tamping. The best answer to these objections is the successful 
use of these joints on several heavy-traffic roads 

^' Bridge "-joints are similar to suspended joints in that the 
joint is supported on two ties, but there is the important differ- 
ence that the bridge joint supports the rail from underneath and 



252 RAILROAD CONSTRUCTION. § 2-11. 

there is no transverse stress in the rail, whereas the suspended 
joint requires the combined transverse strength of both angle- 
bars and rail. A serious objection to bridge-joints lies in the 
fact of their considerable thickness between the rail base and the 
tie. When joints are placed ''staggered" rather than ''oppo- 
site'^ (as is now the invariable standard practice), the ties sup- 
porting a bridge- joint must either be notched down, thus weak- 
ening the tie and promoting decay at the cut, or else the tie 
must be laid on a slope and the joint and the opposite rail do not 
get a fair bearing. 

241. Failures of rail-joints. It has been observed on double- 
track roads that the maximum rail wear occurs a few inches 
beyond the rail gap at the joint in the direction of the traffic. 
On single-track roads the maximum rail wear is found a few 
inches each side of the joint rather than at the extreme ends of 
the rail, thus showing that the rail end deflects down under the 
wheel until (with fast trains especially) the wheel actually jumps 
the space and strikes the rail a few inches beyond the joint, the 
impact producing excessive wear. This action, w^hich is called 
the "drop," is apt to cause the first tie beyond the joint to 
become depressed, and unless this tie is carefully watched and 
maintained at its proper level, the stresses in the angle-bar may 
actually become reversed and the bar may break at the top. The 
angle-bars of a suspended joint are normally in compression at 
the top. The mere reversal of the stresses would cause the bars 



I 



Fig. 118. — Effect of "Wheel Drop" (Exaggerated). 

to give way with a less stress than if the stress were always the 
same in kind. A supported joint, and especially a three-tie 
joint (see § 240), is apt to be broken in the same manner. 

242. Standard angle-bars. An angle-bar must be so made 
as to closely fit the rails. The great multiplicity in the designs 
of rails (referred to in Chapter IX) results in nearly as great 
variety in the detailed dimensions of the angle-bars. The sec- 
tions here illustrated must be considered only as types of the 
variable forms necessary for each different shape of rail. The 



§243. 



RAIL-FASTENINGS; 



253 



absolutely essential features required for a fit are (1) the angles 
of the upper and lower surfaces of the bar where they fit against 
the rail, and (2) the height of the bar. The bolt-holes in the 
bar and rail must also correspond. The holes in the angle-plates 
are elongated or made oval, so that the track-bolts, which are 




^^ 



i" (^^% ^J^'s—gl 



:l_@ 6^^-© 9^^ ®-^ 




Fig. 119. — Standard Angle-bar— 80-lb. Rail. M, C. R.R. 



made of corresponding shape immediately under the head, will 
not be turned by jarring or vibration. The holes in the rails 
are made of larger diameter (by about i'O than the bolts, so as 
to allow the rail to expand with temperature. 

243. Later designs of rail-joints. In Plate VII are shown 
various designs which are competing for adoption. The most 
prominent of these (judging from the discussion in the conven- 
tion of the Roadmasters Association of America in 1897) are 
the '^Continuous" and the ''Weber." Each of them has been 
very extensively adopted, and where used are universally pre- 
ferred to angle-plates. Nearly all the later designs embody 
more or less directly the principle of the bridge-joint, i.e., sup- 
port the rail from underneath. An experience of several years 
will be required to demonstrate which form of joint best satis- 
fies the somewhat opposed requirements of minimum cost (both 
initial and for maintenance) and minimum wear of rails and 
rolling stock. 



254 RAILROAD CONSTRUCTION § 243. 

243a* Proposed specifications for steel splice-bars. The fol- 
lowing specifications for steel splice-bars were proposed in 1900 
by Committee No. 1, American Section, International Associa- 
tion for Testing Materials. 

1. Steel for splice-bars may be made by the Bessemer or open- 
hearth process. 

2. Steel for splice-bars shall conform to the following limits 
in chemical composition: 



Per cent. 

Carbon shall not exceed 0.15 

Phosphorus shall not exceed 0.10 

Manganese 0.30 to 0.60 



3. Splice-bar steel shall conform to the following physical 
qualities : 

Tensile strength, pounds per square inch 54000 to 64000 

Yield point, pounds per square inch 32000 

Elongation, per cent in eight inches shall not 

be less than 25 

4. (a) A test specimen cut from the head of the splice-bar 
shall bend 180° flat on itself without fracture on the outside 
of the bent portion. 

(h) If preferred the bending test may be made on an un- 
punched splice-bar, which, if necessary, shall be first flattened 
and shall then be bent 180° flat on itself without fracture on 
the outside of the bent portion. 

5. A test specimen of 8-inch gauged length, cut from the head 
of the splice-bar, shall be used to determine the physical proper- 
ties specified in paragraph No. 3. 

6. One tensile specimen shall be taken from the rolled splice- 
bars of each blow or melt, but in case this develops flaws, or 
breaks outside of the middle third of its gauged length, it may 
be discarded and another test specimen submitted therefor. 

7. One test specimen cut from the head of the splice-bar shall 
be taken from a rolled bar ot each blow or melt, or if preferred 
the bending test may be made on an unpunched splice-bar, 
which, if necessary, shall be flattened before testing. The bend- 
ing test may be made by pressure or by blows. 

8. For the purposes of this specification, the yield point shall 



i 



§ 244. RAIL-FASTENINGS. 255 

be determined by the careful observation of the drop of the 
beam or halt in the gauge of the testing machine. 

9. In order to determine if the material conforms to the chem- 
ical limitations prescribed in paragraph No. 2 herein, analysis 
shall be made of drillings taken from a small test ingot. 

10. All splice-bars shall be smoothly rolled and true to templet. 
The bars shall be sheared accuratel}- to length and free from 
fins or cracks, and shall perfectly fit the rails for which they are 
intended. The punching and notching shall accurately conform 
in every respect to the drawing and dimensions furnished. 

11. The name of the maker and the year of manufacture shall 
be rolled in raised letters on the side of the splice-bar. 

12. The inspector representing the purchaser shall have all 
reasonable facilities afforded to him b}^ the manufacturer, to 
satisfy him that the finished material is furnished in accordance 
with these specifications. All tests and inspections shall be 
made at the place of manufacture, prior to shipment. 



TIE-PLATES. 

244. Advantages, (a) As already indicated in § 204, the 
life of a soft-wood tie is very much reduced by "rail-cutting" 
and "spike-killing," such ties frequently requiring renewal long 
before any serious decay has set in. It has been practically 
demonstrated that the "rail-cutting" is not due to the mere 
pressure of the rail on the tie, even with a maximum load on 
the rail, but is due to the impact resulting from vibration and 
to the longitudinal working of the rail. It has been proved 
that this rail-cutting is practically prevented b}^ the use of tie- 
plates, (b) On curves there is a tendency to overturn the outer 
rail due to the lateral pressure on the side of the head. 
This produces a concentrated pressure of the outer edge of the 
base on the tie which produces rail-cutting and also draws the 
inner spikes. Formerly the only method of guarding against 
this was by the use of "rail-braces," one pattern of which is 
sho^Mi in Fig. 120. But it has been found that tie-plates serve 
the purpose even better, and rail-braces have been abandoned 
where tie-plates are used, (c) Driving spikes through holes 
in the j)late enables the spikes on each side of the rail to mutually 
support each other, no matter in which (lateral) direction the 
rail may tend to move, and this probably accounts in large 



256 



RAILROAD CONSTRUCTION, 



§245. 



measure for the added stability obtained by the use of tie-plates 
{(1) The wear in spikes, called ''necking/' caused by the ver- I 
tical vibration of the rail against them, is very greatly reduced, 
(e) The cost is very small compared with the value of the added 
life of the tie, the large reduction in the work of track main- 



i 




I 



Fig. 120. 



tenance, and the smoother running on the better track which is 
obtained. It has been estimated that by the use of tie-plates 
the life of hard- wood ties is increased from one to three years, 
and the life of soft-wood ties is increased from three to six 
years. From the very nature of the case, the value of tie-plates 
is greater when they are used to protect soft ties. 

245. Elements of the design. The earliest forms of tie-plates 
were flat on the bottom, but it was soon found that they would 
work loose, allow sand and dirt to get between the rail and the 
plate and also between the plate and the tie, which would cause 
excessive wear. Such plates are also apt to produce an objec- 
tionable rattle Another fault of the earlier designs was the use 
of plates so thin that they would buckle. The latest designs 
have flanges of ''teeth" formed on the lower surface which 
penetrate the tie about f to If". Opinion is still divided on 
the question of whether these teeth should run with the grain 
or across the grain. If the flanges run with the grain, they 
generally extend the whole length of the tie-plate — as in the 
Wolhaupter design. If the grain is to be cut crosswise, several 
teeth about 1" wide will be used — as in the Goldie design. 

It is a ver\' important feature that the spike -holes should be 
so punched that the spikes will fit closely to the base of the rail. 
Otherwise a lateral motion of the rail will be permitted which 
will defeat one of the main objects of the use of the plate. 



PLATE VII 




WEBER RAIL JOINT. 





3NZAN0 JOINT. 




(To face page 25G ) 



246, 



RAIL-FASTENINGS, 



257 



Another unsettled detail is the use of "shoulders^' on the 
upper surface. On the one hand it is claimed that the use of 
shoulders relieves the spikes of side pressure from the rail and 
prevents '' necking*'^ On the other hand it is claimed that if the 




GOLDIE 

Fig. 121. — Tie-plates. 



the plain plate is once properly set with new spikes (at least vnth 
spikes not already necked) the spikes will not neck appreci- 
ably, and that, as the shouldered plates cost more, the additional 
expenditure is unnecessary. 

The above designs should be studied with reference to the 
manner in which they fulfill the requirements which have been 
already stated. As in the ease of rail- joints, the best forms of 
tie-plates are of comparatively recent design, and experience with 
them is still insufficient to determine beyond all question which 
designs are the best. 

246. Method of setting. A very important detail in the 
process of setting the tie-plates on the ties is that the flanges or 
teeth should penetrate the tie as far as desired w^hen the plates 
are first put in position. It requires considerable force to press 
the teeth into a tie. In a few cases trackmen have depended on 
the easy process of waiting for passing trains to force the teeth 



258 



RAILROAD CONSTRUCTION. 



§247 



down. Until the teeth are down the spikes cannot be driven 
home, and this apparently cheap and easy process results in loose 
spikes and rails. If the trackmen neglect even temporarily to 
tighten these spikes, it will become impossible to make them 
tight ultimately. The plates are generally pounded into place 
with a 10- to 16-pound sledge-hammer. A very good method 
was adopted once during the construction of a bridge when a 
pile-driver was at hand. The bridge-ties were placed under the 
pile-hammer. The plates, accurately set to gauge, w^ere then 
forced in by a blow from the 3000-lb. hammer falling 2 or 3 feet 



SPIKES. 



247. Requirements. The rails must be held to the ties by a 
fastening which will not only give sufficient resistance, but which 
■will retain its capacity for resistance. It must also be cheap 
and easily applied. The ordinary track-spike fulfills the last 
requirements, but has comparatively small resisting power, com- 
pared with screws or bolts. Worse than all, the tendency to 




;^ 



Fig. 122. 





Fig. 123. 



vertical vibration in the rail produces a series of upward pulls on 
the spike that soon loosens it. When motion has once begun 
the capacity for resistance is greatly reduced, and but little more 
vibration is required to pull the spike out so much that redriving 
is necessary. Driving the spike to place again in the same hole 



§248 



RAIL-FASTENINGS. 



259 



is of small value except as a very temporary expedient, as its 
holding power is then very small, Redriving the spikes in new 
holes very soon ''spike-kills" the tie. Many plans have been 
devised to increase the holding power of spikes, such as making 
them jagged, twisting the spike, swelling the spike at about the 
center of its length, etc. But it has been easily demonstrated 
that the fibers of the wood are generally so crushed and torn by 
driving such spikes that their holding power is less than that of 
the plain spike. 

The ordinary spike (see Fig. 122) is made with a square cross- 
section which is uniform through the middle of its length, the 
lower 1 J" tapering down to a chisel edge, the upper part swelling 
out to the head. The Goldie spike (see Fig. 123) aims to im- 
prove this form by reducing to a minimum the destruction of the 
fibers. To this end, the sides are made smooth, the edges are 
clean-cut, and the point, instead of being chisel-shaped, is ground 
down to a pyramidal form. Such fiber-cutting as occurs is thus 
accomplished without much crushing, and the fibers are thus 
pressed away from the spike and slightly do^\^lward. Any 
tendency to draw the spike will therefore cause the fibers to 
press still harder on the spike and thus increase the resistance. 

248. Driving. The holding power of a spike depends largely 
on how it is driven. If the blows 
are eccentric and irregular in direc- 
tion, the hole will be somewhat en" 
larged and the holding power largely 
decreased. The spikes on each 
side of the rail in an}^ one tie should 
not be directly opposite, but should 
be staggered Placing them direct- 
ly opposite will tend to split the tie, 
or at least decrease the holding 
power of the spikes. The direction 
of staggering should be reversed in 
the two pairs of spikes in any one 
tie (see Fig. 124). This will tend to prevent any twisting of the 
tie in the ballast, which would otherwise loosen the rail from the 
tie. 

249. Screws and bolts. The use of these abroad is very ex- 
tensive, but their use in this country has not passed the experi- 
jnental stage. The screws are ''wood "-screws (see Fig. 125), 



>'' 



Fig. 124. — Spike-dkiving. 



260 



RAILROAD CONSTRUCTION. 



§ 250. 



having large square heads, which are screwed down with a track- 
wrench. Holes, having the sanie diameter as the base of the 
screw-heads, should be first bored into the tie, at exactly the 
right position and at the proper angle with the vertical. A 
light wooden frame is sometimes used to guide the auger at the 
proper angle. Sometimes the large head of the screw bears 
directly against the base of the rail, as with the ordinary spike. 
Other designs employ a plate, made to fit the rail on one side, 
bearing on the tie on the other side, and through which the screw 
passes. These screws cost much more than the spikes and re- 
quire more work to put in place, but their holding poAver is much 
greater and the work of track maintenance is very much less. 
Screw-bolts, passing entirely through the tie, having the head 
at the bottom of the tie and the nut on the upper side, are also 
used abroad. These are quite difficult to replace, requiring that 
the ballast be dug out beneath the tie, but on the other hand the 





Fig. 125. 



Fig. 126. 



occasions for replacing such a bolt are comparatively rare, as 
their durability is very great. The use of screws or bolts in- 
creases the life of the tie by the avoidance of ^' spike-killing." It 
is capable of demonstration that the reduced cost of mainte- 
nance and the resulting improvement in track would much more 
than repay the added cost of screws and bolts, but it seems im- 
possible to induce railroad directors to authorize a large and 
immediate additional expenditure to make an annual saving 
whose value, although unquestionably considerable, cannot be 
exactly computed. 

250. "Wooden spikes." Among the regulations for track- 
laying given in § 208, mention was made of wooden ^' spikes/' 



§251. 



RAIL-FASTENINGS. 



261 



or plugs, which are used to fill up the holes when spikes are 
withdrawn. The value of the policy of filling up these holes is 
unquestionable, since the expense is insignificant compared with 
the loss due to the quick and certain decay of the tie if these 
holes are allowed to fill with water and remain so. But the 
method of making these plugs is variable. On some roads they 
are " hand-made '^ by the trackmen out of otherwise use- 
less scraps of lumber, the work being done at odd mo- 
ments. This policy, while apparently cheap, is not 
necessarily so, for the hand-made plugs are irregular 
in size and therefore more or less inefficient. It is 
also quite probable that if the trackmen are required to 
make their own plugs, they would spend time on these 
ver}^ cheap articles which could be more profitably em- 
plo3^ed otherwise. Since the holes made by the spikes 
are larger at the top than they are near the bottom, the 
plugs should not be of uniform cross-section but should 
be slightly wedge-shaped. The ^^Goldie tie-plug'' 
(see Fig. 127) has been designed to fill these require- 
ments. Being machine-made, they are uniform in 
size; they are of a shape which will best fit the hole; 
they can be furnished of any desired wood, and at a 
cost which makes it a wasteful economy to attempt 
to cut them by hand. 




Fig. 127. 



TRACK-BOLTS AND NUT- LOCKS. 

251. Essential requirements. The track-bolts must have 
sufficient strength and must be screwed up tight enough to hold 
the angle-plates against the rail with sufficient force to develop 
the full transverse strength of the angle-bars. On the other 
hand the bolts should not be screwed so tight that slipping may 
not take place when the rail expands or contracts with tempera- 
ture. It would be impossible to screw the bolts tight enough to 
prevent slipping during the contraction due to a considerable fall 
of temperature on a straight track, but when the track is curved, 
or when expansion takes place, it is conceivable that the resist- 
ance of the ties in the ballast to lateral motion may be less than 
the resistance at the joint. A test to determine this resistance 
was made by Mr. A. Torrey, chief engineer of the Mich. Cent. 
R. R., using 80-lb. rails and ordinary angle-bars, the bolts being 
screwed up as usual. If required a force of about 31000 to 



262 



RAILROAD CONSTRUCTIOX. 



§ 252. 



35000 lbs. to start the joint, which would be equivalent to the 
stress induced by a change of temperature of about 22°. But 
if the central angle of any given curve is small, a comparatively 
small lateral component will be sufficient to resist a compression 
of even 35000 lbs. in the rails. Therefore there will ordinarily 
be no trouble about having the joints screwed too tight. The 
vibration caused by the passage of a train reduces the resistance 
to slipping. This vibration also facilitates an objectionable 
feature, viz., loosening of the nuts of the track-bolts. The bolt 
is readily prevented from turning by giving it a form which is 
not circular immediately under the head and making corre- 
sponding holes in the angle-plate. Square holes would answer 
the purpose, except that the square corners in the holes in the 
angle-plates would increase the danger of fracture of the plates. 
Therefore the holes (and also the bolts, under the head) are 
made of an oval form, or perhaps a square form with rounded 
corners, avoiding angles in the outline. 

The nut-locks should be simple and cheap, should have a life 
at least as long as the bolt, should be effective, and should not 
lose their effectiveness wdth age. Many of the designs that have 
been tried have been failures in one or more of these particulars 
as will be described in detail below. 

252. Design of track-bolts. In Fig. 128 is shown a common 
design of track-bolt. In its general form this represents the 

bolt used on nearly all roads, 
being used not only with the 
common angle-plates, but also 
with man}^ of the improved de- 
signs of rail-joints. The varia- 
tions are chiefly a general in- 
crease in size to correspond with 
the increased weight of rails, 
besides variations in detail di- 
mensions which are frequently 
unimportant. The diameter is 
usually f" to Y'; l" bolts are 
sometimes used for the heaviest 
sections of rails. As to length, 

the bolt should not extend more 
Fig. 128.-TRACK-BOI.T. ^^^^ ^„ ^^^^.^^ ^^ ^^^ ^^^ ^^.j^^^ 

it is screT>^ed up. If it extends farther than this it is liable to be 




§ 253. RAIL-FASTENINGS. 263 

broken off by a possible derailment at that point. The lengths 
used vary from 3|", which may be used with 60-lb. rails, to 5", 
which is required with 100-lb. rails. The length required de- 
pends somewhat on the type of nut-lock used. 

253. Design of nut-locks. The designs for nut-locks may be 
divided into three classes: (a) those depending entirely on an 
elastic washer which absorbs the vibration which might other- 
wise induce turning; (h) those which jam the threads of the 
bolt and nut so that, when screwed up, the frictional resistance 
is too great to be overcome by vibration; (c) the '' positive'* 
nut-locks — those which mechanically hold the nut from turning. 
Some of the designs combine these principles to some extent. 
The ^'vulcanized fiber" nut-lock is an example of the first class. 
It consists essentially of a rubber washer which is protected by 
an iron ring. When first placed this lock is effective, but the 
rubber soon hardens and loses its elasticity and it is then ineffec- 
tive and worthless. Another illustration of class (a) is the use 
of wooden blocks, generally 1'' to 2'' oak, which extend the 
entire length of the angle-bar, a single piece forming the washer 
for the four or six bolts of a joint. This form is cheap, but the 
wood soon shrinks, loses its elasticity, or decays so that it soon 
becomes worthless, and it requires constant adjustment to keep 
it in even tolerable condition. The '' Verona '* nut-lock is 
another illustration of class (a) which also combines some of the 
positive elements of class (c). It is made of tempered steel and, 
as shown in Fig. 129, is warped and has sharp edges or points. 
The warped form furnishes the element of elastic pressure when 
the nut is screwed up. The steel being harder than the iron of 
the angle-bar or of the nut, it bites into them, owing to the 
great pressure that must exist when the washer is squeezed 
nearly flat, and thus prevents any backward movement, although 
forward movement (or tightening the bolt) is not interfered 
with. The " National'* nut-lock is a type of the second class (6), 
in which, like the ^' Harvey" nut-lock, the nut and lock are com- 
bined in one piece. With six-bolt angle-bars and 30-foot rails, 
this means a saving of 2112 pieces on each mile of single track. 
The "National" nuts are open on one side. The hole is drilled 
and the thread is cut slightly smaller than the bolt, so that when 
the nut is screwed up it is forced slightly open and therefore 
presses on the threads of the bolt with such force that vibration 
cannot jar it loose. Unlike the " National" nut, the ^' Harvey " 



264 



RAILROAD CONSTRUCTION. 



§ 253. 



nut is solid, but the form of the thread is progressively varied so 
that the thread pinches the thread of the bolt and the frictional 
resistance to turning is too great to be affected b}^ vibration. 

The "Jones'' nut-lock, belonging to class (c), is a type of a 
nut-lock that does not depend on elasticity or jamming of screw- 
threads. It is made of a thin flexible plate, the square part of 




VERONA 





VULCANIZED FIBRE 



f 


C^5 


n 


\ 


' — 1 


J 



NATIONAL 



\\u^^ 




JONES 
EXCELSIOR 

Fig. 129. — Types of Nut-locks. 



which is so large that it will not tiu-n after being placed on the 
bolt. After the nut is screwed up, the thin plate is bent over so 
that the re-entrant angle of the plate engages the corner of the 
nut and thus mechanically prevents any turning. The metal 
is supposed to be sufficiently tough to endure without fracture 
as many bendings of the plate as will ever be desired. Nut- 
locks of class (c) are not in common use. 



CHAPTER XI. 

SWITCHES AND CROSSINGS. 
SWITCH CONSTRUCTION. 

254. Essential elements of a switch. Flanges of some sort are 
a necessity to prevent car- wheels from running off from the rails 
on which they may be moving. But the flanges, although a 
necessity, are also a source of complication in that they require 
some special mechanism which will, when desired, guide the 
wheels out from the controlling influence of the main-line rails. 
This must either be done by raising the wheels high enough 
so that the flanges may pass over the rails, or by breaking the 
continuit}^ of the rails in such a wa}^ that channels or "flange 
spaces'' are formed through the rails. An ordinary stub-switch 
breaks the continuity of the main-line rails in three places, two 
of them at the switch-block and one at the frog. The Wharton 
switch avoids two of these breaks by so placing inclined planes 
that the wheels, rolling on their flanges, will surmount these 
inclines until they are a little higher than the rails. Then the 
wheels on the side toward which the switch runs are guided 
over and across the main rail on that side This rise being ac- 
complished in a short distance, it becomes impracticable to 
operate these switches except at slow speeds, as any sudden 
change in the path of the center of gra\'ity of a car causes very 
destructive jars both to the ST^itch and to the roUmg stock. The 
other general method makes a break in one main rail (or both) 
at the switch-block. In both methods the wheels are led to one 
side by means of the ^4ead rails," and finally one line of wheels 
passes through the main rail on that side by means of a "frog." 
There are some designs by which even this break in the main 
rail is avoided, the wheels being led over the main rail by means 
of a short ynovahle rail which is on occasion placed across the 
main rail, but such designs have not come into general use. 

255. Frogs. Frogs are provided with two channel- ways or 
"flange spaces" through which the flanges of the wheels move, 

265 



266 



RAILROAD CONSTRUCTION. 



§ 255. 



Each channel cuts out a parallelogram from the tread area. 
Since the wheel-tread is always wider than the rail, the wing 
rails will support the wheel not only across the space cut out by 
the channel, but also until the tread has passed the point of the 
frog and can obtain a broad area of contact on the tongue of the 
frog. This is the theoretical idea, but it is very imperfectly 




•♦—TOWARD SWITCH 



Fig. 130. — Diagrammatic Design of Frog. 



realized. The wing rails are sometimes subjected to excessive 
wear owing to ^'hollow treads'^ on the wheels — owing also to 
the frog being so flexible that the point ^' ducks" when the wheel 
approaches it. On the other hand the sharp point of the frog 
will sometimes cause destructive wear on the tread of the wheel. 
Therefore the tongue of the frog is not carried out to the sharp 
theoretical point, but is purposely somewhat blunted. But 
the break which these channels make in the continuity of the 
tread area becomes extremely objectionable at high speeds, 
being mutually destructive to the rolling stock and to the frog. 
The jarring has been materially reduced by the device of ^' spring 
frogs" — to be described later. Frogs were originally made of 
cast iron — then of cast iron with wearing parts of cast steel, 
which were fitted into suitable notches in the cast iron. This 
form proved extremely heavy and devoid of that elasticity of 
track which is necessary for the safety of rolling stock and 
track at high speeds. The present universal practice is to build 
the frog up of pieces of rails which are cut or bent as required. 
These pieces of rails (at least four) are sometimes assembled by 
riveting them to a flat plate, but this method is now but little 
used, except for very light work. The usual practice is now 
chiefly divided between ''bolted" and ''keyed" frogs. In each 
case the space between the rails, except a sufficient flange-way, 
is filled with a cast-iron filler and the whole assemblage of parts 






i 



§ 256 SWITCHES AND CROSSINGS. 267 

is suitably bolted or clamped together, as is illustrated in Plate 
VIII. The operation of a spring-rail frog is evident from the 
figure. Since a siding is usually operated at slow speed, while 
the main track may be operated at fast speed, a spring-rail frog 
^vill be so set that the tread is continuous for the main track and 
broken for the siding. This also means that the spring-rail will 
only be moved by trains moving at a (presumably) slow speed 
on to the siding. For the fast trains on the main line such a 
frog is substantially a ^' fixed'' frog and has a tread which is 
practically continuous. 

256. To find the frog number. The frog number (n) equals 
the ratio of the distance of an}^ part on the tongue of the frog 
from the theoretical point of the frog divided by the width of 
the tongue at that point, i.e. =hc^ah (Fig. 130). This value 
may be directly measured by applying any convenient unit of 
measure (even a knife, a short pencil, etc.) to some point of the 
tongue where the width just equals the unit of measure, and then 
noting how many times the unit of measure is contained in the 
distance from that place to the theoretical point. But since c, 
the theoretical point, is not so readil}^ determinable with exacti- 
tude, it being the imaginary intersection of the gauge lines, it 
may be more accurate to measure de, ah, and hs; then n, the frog 
number, =hs-^(ah + de) . If the frog angle be called F, then 

n=hc^ab=hs-^(ah-\-de) =,^ cot ^F; 
i.e., cot iF = 2n, 

257. Stub switches. The use of these, although once nearly 
universal, has been practically abandoned as turnouts from 
main track except for the poorest and cheapest roads. In some 
States their use on main track is prohibited by law. They have 
the sole merit of cheapness with adaptability to the circum- 
stances of very light traffic operated at slow speed when a con- 
siderable element of danger may be tolerated for the sake of 
economy. The rails from A to B (see Fig. 131 *) are not fastened 

* The student should at once appreciate that in Fig. 131, as well as in 
nearly all the remaining figures in this chapter, it becomes necessary to use 
excessively large frog angles, short radii, and a very wide gauge in order to 
illustrate the desired principles with figures which are sufficiently small for 
the page. In fact, the proportions used in the figures are such that serious 
mechanical difficulties would be encountered if they were used. These dif- 
ficulties are here ignored because the^' can be neglected in the proportions 
used in practice, 




(Jo face page 267.) 



268 



RAILROAD CONSTRUCTION. 



§ 257. 



to the ties; they are fastened to each other by tie-rods which 
keep them at the proper gauge; at and back of B they are \ 




Fig. 131. — Stub Switch. 

securely spiked to the ties, and at A they are kept in place by 
the connecting bar (C) fastened to the switch-stand. One great 
objection to the switch is that, in its usual form, when operated 
as a trailing switch, a derailment is inevitable if the switch is 
misplaced. The very least damage resulting from such a derail- 
ment must include the bending or breaking of the tie-rods of the 
switch-rail. Several devices have been invented to obviate this 
objection, some of Avhich succeed very well mechanically, al- 
though their added cost precludes any economy in the total cost 
of the switch. Another objection to the switch is the looseness 
of construction which makes the switches objectionable at high 
speeds. The gap of the rails at the head-block is always con- 
siderable, and is sometimes as much as two inches. A driving- 




FiG. 132. — Point Switch. 



wheel with a load of 12000 to 20000 pounds, jumping this gap 
-^ith any considerable velocity, will do immense damage to the 



§ 258. 



SWITCHES AND CROSSINGS. 



269 



j farther rail end, besides producing such a stress in the construc- 
I tion that a breakage is rendered quite likely, and such a breakage 
might have very serious consequences. 

258. Point switches. The essential principle of a point switch 
is illustrated in Fig. 132. As is shown, one main rail and also 
one of the switch-rails is unbroken and immovable. The other 
■ main rail (from A to F) and the corresponding portion of the 
' other lead rail are su]:)stantia]ly the same as in a stub switch. 
I A portion of the main rail (AB) and an equal length of the oppo- 
' site lead rail (usually 15 to 24 feet long) are fastened together" 
by tie-rods. The end at A is jointed as usual and the other end 
[ is pointed, both sides being trimmed down so that the feather 
edge at B includes the web of the rail. In order to retain in it 
, as much strength as possible, the point-rail 
is raised so that it rests on the base of the 
stock-rail, one side of the base of the 
point-rail being entirely cut away. As 
may be seen in Fig. 133, although the in- 
fluence of the point of the rail in moving 
the wheel-flange away from the stock-rail 
is really zero at that point, yet the rail has 
all the strength of the web and about one- 
half that of the base — a very fair angle- 
iron. The planing runs back in straight 
lines, until at about six or seven feet back 
from the point the full width of the head is 
obtained. The full width of the base will only be obtained at 
about 13 feet from the point. An 80-1 b. rail is 5 inches wdde at 
the base. Allowing f more for a spike between the rails, this 
gives 5f" as the minimum width between rail centers at the 
joint. The minimum angle of the switch-point (using a 15-foot 

5 75 

point-rail) is therefore the angle whose tangent is — ' = 

i-D /\ I^ 

.03914, w^hich is the tangent of 1° 50'. S^^dtch-raiis are some- 
times used with a length of 24 feet, which reduces the angle of 
the swdtch-point to 1° 09'. 




Ftg. 133. 



259. Switch-stands. The simplest and cheapest form is the 
"ground lever," which has no target. The radius of the circle 
described b}^ the connecting-rod pin is precisely one-half the 
throw. From the nature of the motion the device is practically 



270 



RAILROAD CONSTRUCTION. 



§ 260. 



self-locking in either position, padlocks being only used to pre- 
vent malicious tampering. The numerous designs of upright' 
stands are always combined with targets, one design of which is 




Fig. 134, — Ground Lever for Throwing 
A Switch. 



cOl^' 



Fig. 135, 



illustrated in Fig. l.?5. When the road is equipped with inter- 
locking signals, the switch-throw mechanism forms a part of the 
design 

260. Tie-rods. These are fastened to the webs of the rails by 
means of lugs which are bolted on, there being usually a hinge- 
joirit between the rod and the lug. Four such tie-rods are 



j § 261 



SWITCHES AND CROSSINGS. 



271 



generally necessary. The first rod is sometimes made with- 
' out hinges, which gives additional stiffness to the comparatively 
' weak rail-points. The old-fashioned tie-rod, having jaws 
i fitting the base of the rail, was almost imiversally used in the 
days of stub switches. One great inconvenience in their use 
lies in the fact that they must be slipped on, one by one, over 
ll the free ends of the sw^itch-rails. Sometimes the lugs are 
\ fastened to the rail-webs by rivets instead of bolts. 




261. Guard-rails. As shown in Figs. 131 and 132, guard-rails 

arc used on both tlie main and switch tracks opposite the frog- 

t point. Their funcrion is not only to prevent the possibility of 



272 



RAILROAD CONSTRUCTION. 



§ 262. 



the wheel-flanges passing on the wrong side of the frog-point, 
but also to save the side of the frog-tongue from excessive wear. 
The necessity for their use may be realized by noting the apparent 
wear usually found on the side of the head of the guard-rail. 
The flange-way space between the heads of the guard-rail and 
wheel-rail therefore becomes a definite quantity and should equal 
about two inches. Since this is less than the space between 
the heads of ordinary (say 80-pound) rails when placed base to 
base, to say nothing of the f nccessar}^ for spikes, it becomes 
necessary to cut the flange of the guard-rail. The length of the 
rail is made from 10 to 15 feet, the ends being bent as shown 
in Fig. 132, so as to prevent the possibility of the end of the 
rail being struck by a wheel-flange. 



MATHEMATICAL DESIGN OF SWITCHES. 

In all of the following demonstrations regarding switches, 
turnouts, and crossovers, the lines are assumed to represent the 
gauge-lines — i.e., the lines of the inside of the head of the rails. 
262. Design with circular lead-rails. The simplest method 

is to consider that the lead-rails 
curve out from the main track- 
rails by arcs of circles which are 
tangent to the main rails and 
which extend to the frog-point F. 
The simple curve from D to F is 
of such radius that (r + ig) vers F 
=g, in which i^ = the frog angle, 
g= gauge, L=the ^4ead^' (BF), 
and r = the radius of the center of 
the switch-rails. 




Fig. 137. 



r+ig 



vers F' 



(74) 



Also, 
Also, 



BF^BD^cotiF; BD=g; BF =L, 

.-. L=^gQot^F (75) 

L = {r-\-lg)smF\ ..... (76) 

QT = 2rs\nhF (77) 



These formulae involve the angle F. As shown in Table ITT, 
the angles {F) are always odd quantities, and their trigonometric 
functions are somewhat troublesome to obtain closely with 



§ 262. SWITCHES AND CROSSINGS. 273 

ordinary tables. The formuLne may be simplified by substitut- 
ing the frog-number n, from the relation that n = JcotJF. 
Since 

r — \g=LQotF and r + i^=i^ cosec i^, 

then r = iL (cot F -\- cosec F) 

= ig ^ot JF(cot i^ + cosec F) 

= hg cot^ iFj since (cot a + cosec a) =cot Ja 

i=2gn^ (78) 

Also, L = 2gn, (79) 

from which r = nXL (80) 

These extremely simple relations may obviate altogether the 
necessity for tables, since they involve only the frog-number and 
the gauge. On account of the great simplicity of these rules, 
they are frequently used as they are, regardless of the fact that 
the curve is never a uniform simple curve from switch- block to 
frog. In the first place there is a considerable length of the 
gauge-line within the frog, which is straight unless it is pur- 
posely curved to the proper curve while being manufactured, 
which is seldom if ever done — except for the very large-angled 
frogs used for street-railway work, etc. It is also doubtful whether 
the switch-rails {BA, Fig. 181) are bent to the computed curve 
when the rails are set for the switch. The switch-rails of point 
switches are .straight, thus introducing a stretch of straight track 
which is about one-fifth of the total length of the lead-rails. The 
effect of these modifications on the length and radius of the lead- 
rails will be developed and discussed in the next four sections. 

The throw (t) of a stub switch depends on the weight of the 
rail, or rather on the width of its base. The throw must be at 
least J" more than that width. The head-block should there- 
fore be placed at such a distance from the heel of the switch (B) 
that the versed sine of the arc equals the throw. These points 
must be opposite on the two rails, but the points on the two rails 
where these relations are exactly true w^ill not be opposite. 
Therefore, instead of considering either of the two radii (r + hg) 
and (r — ^g), the mean radius r is used. Then (see Fig. 137) 

vers /vOQ=/--r, 

and the length of the switch-rails is 

QK=r sin KOQ (81) 



274 



RAILROAD CONSTRUCTION. 



§ 263. 



These relations develop another disadvantage in the use of a 
stub switch. The required value of BG, using a No. 10 frog 
and 80-pound rail, is 30.1 feet — slightly more than a full rail 
length. It would be unsafe to leave so much of the track un- 
spiked from the ties. Whether this is obviated by spiking down 
a portion of the switch-rails (virtually shortening the lead) or by 
moving the sw^itch-block nearer the heel of the switch (shorten- 
ing the switch-rails), but still maintaining the required throw, 
the theoretical accuracy of the curve is hopelessly lost. 

263. Effect of straight frog-rails. A portion of the ends of 

the rails of a frog are free and may 
be bent to conform to the switch- 
rail curve, but there is a consid- 
erable portion which is fitted to 
the cast-iron filler, and this por- 
tion is always straight. Call the 
length of this straight portion 
back from the frog-point / ( =FH, 
Fig. 138). Then we have 
r-^ig = (g-f sin F) --vers F 

=-^-/cotii^ 
vers F ' 



^ 




g 



Fig. 138. = ^r 

vers F 

BF=L = (g-f sin F) cotiF+fcosF 

=2gn-f sin F cot iF-i-f cos F 

=2gn-f(l-\~cosF)-}-fcosF 

-=2gn-f 



2/n. 



(82) 



(83) 



Since r—^g = {L—f sec F) cot F, and 
r^y = (L—f cos F) cosec F, 
r=iL(cot i^ + cosec F) —J/ sec F cot F — iJ cos F cosec F 

-^-»'(w-')- 



r=Ln-if cot iF 

= Ln -fn. Then from (83) 
r=2gn^—2fn 



. (84) 



264. Effect of straight point-rails. The "point switches,'' 
now so generally used, have straight switch-rails. This requires 



^ 



§ 265. 



SWITCHES AND CROSSINGS. 



275 



an angle in the alignment rather than turning off by a tangential 
curve. The angle is, however, very small (between 1° and 2°), 
and the disadvantages of this angle are small compared with 
the very great advantages of the device. 




-a 



2 sin i(i^ + a) sin h{F-a) 
g—k 

= 77T* * • • * 

COS a — cos i^ 

BF^L= FM cos i(F f a) + DN 

= {g- k) cot KF + a) + Z)i\r. . 



(85) 



(86) 



I 



265. Combined effect of straight frog-rails and straight point- 
rails. It becomes necessary in this case to find a curve which 
shall be tangent to both the point-rail and the frog-rail. The 
curve therefore begins at M, its tangent making an angle of a 
(usually 1° 50') with the main rail, and runs to H. The central 



276 



RAILROAD CONSTRUCTION. 



260 



angle of the curve is therefore (F — a). The angle of the chord 
HM with the main rails is therefore 



HM== 



g—f sin F—k ^ 



r+ig== 



smi(F + a) ' 

HM 

2sini(F-a) 

g—f sin F—k 
' 2 sin i(F + a) sin i(F-a) 

g—f sin F — k^ 



cos a — COS F 



ST =2r sin i(F- a). 



BF=L=HM cos i(F + a) +/ cos F+DN 

= (g-f sin F-k) cot K^ + a) +/ cos F+DN. 



(87) 
(88) 

(89) 



i 



It may be more simple, if (r + ^g) has already been computed, 
to write 

L =2(r + ig) sin i(^-a) cos i(F-\-a) +/ cos F+DN 
= (r + ig)(sin F -sin a) -{-f cos F+DN (90) 



.-" 



/'I' 



" MN=fc 
. FH=/ 

-^F-a VHMR=M(F-a) 

1 




Fig. 140. 



266. Comparison of the above methods. Computing values 
for r and L by the various methods, on the uniform basis of a 



i 



§267. 



SWITCHES AND CROSSINGS. 



277 



No. 9 frog, standard gauge 4' 8y\ /=3'.37, /j=5f"=0'.479, 
DN = 15' 0", and a = l° 50', we may tabulate the comparative 
results: 



Deg. of curve 
L 



§262. 
Simple circle. 
Curved frog- 
rail. Curved 
switch-rail. 



762.75 
7° 31' 
84.75 



§263. 

Straight 

frog-rail. 

Curved 

switch-rail. 



702.00 
8° 10' 
81.37 



§264. 
Curved frog- 
rail. Straight 
switch-rail. 



747.48 
7° 40' 
74.00 



§265. 

Straight 

frog-rail. 

Straight 

switch-rail. 



681 . 16 
8° 25' 
72.13 



This shows that the effect of using straight frog-rails and 
straight switch-rails is to sharpen the curve and shorten the lead 
in each case separately, and that the combined effect is still 
greater. The effect of the straight switch-rails is especially 
marked in reducing the length of lead, and therefore Eq. 78 to 
80, although having the advantage of extreme simplicity, can- 
not be used for point-switches without material error. The 
effect of the straight frog-rail is less, and since it can be mate- 
rially reduced by bending the free end of the frog-rails, the in- 
fluence of this feature is frequently ignored, the frog-rails are 
assumed to be curved, and Eq. 85 and 86 are used. (See § 276 
for a further discussion of this point.) 

2^7. Dimensions for a turnout from the OUTER side of a curved 




track. In this demonstration the switch-rails will ,be considered 
as uniformly circular from the switch-points to the frog-point. 



278 RAILROAD CONSTRUCTION. § 267. 

In the triangle FCD (Fig. 141) we have 
{FC-^CD) :(FC -CD) y.tan i(FDC + DFC) '.tan i(FDC-DFC); 
but i(FDC + DFO = 90° - id 

and i(FDC-DFC)=iF. 

Also, FC-^CD=2R and FC-CD=g; 

.*. 2R :g y cot ^6 : tan Ji^ 
:: cot Ji^ : tan J/?; 



.-. tani^ = ^ (91) 

Also, OF : FC :: sin 6 : sin 0; but <}) = (F-d); 



then ^ + *^ = (^ + i^)sW^ (^2> 

5i^=7, = 2(i^ + ig)sinJ^ (93) 

Tf the curvature of the main track is very sharp or the frog 
angle unusually small, F may be less than 6 ; in which case the 
center will be on the same side of the main track as C. Eq. 
92 will become (by calling r= —r and changing the signs) 

(._i,)=(J2 + i,^^._^;-l- (94) 

If we call d the degree of curve corresponding to the radius 
r, and D the degree of curve corresponding to the radius R, also 
d' the degree of curve of a turnout from a straight track (the frog 
angle F being the same), it may be shown that d = d' —D (very 
nearly). To illustrate we will take three cases, a number 6 
Irog (very blunt), a number 9 frog (very commonly used), and a 
number 12 frog (unusually sharp). Suppose D=4° 0'; also 
i) = 10° 0'; g=4' 8J''=4'.7b8. 

A brief study of the tabular form on p. 279 will show that the 
error involved in the use of the approximate rule for ordinary 
curves (4° or less) and for the usual frogs (about No. 9) is really 
insignificant, and that, even for sharper curves (10° or more), 
or for very blunt frogs, the error would never cause damage, 
considering the lower probable speed. In the most imfavorable 
case noted above the change in radius is about 1%. On account 
of the closeness of the approximation the method is frequentl}^ 
used. The remarkable agreement of the computed values of L 



i 
I 



§ 268, 



SWITCHES AND CROSSINGS. 



279 



Frog 




Z) = 4°. 




"L" for 


num- 
ber. 


d 


d'-D 


Error. 


L 


straight 
track. 


6 

9 

12 


12° 54' 20" 
3 30 27 
13 33 


12° 57' 52" 
3 31 04 
13 36 


0° 03' 32" 
37 
03 


56.57 

84.85 
112.72 


56.50 

84.75 

113.00 



Frog 


Z) = 10°. 


"L" for 
straight 
track. 


num- 
ber. 


d 


d'-D 


Error. 


L 


6 

9 

12 


6° 53' 24" 
2 27 54 
5 44 26 


6° 57' 52" 
2 28 56 
5 46 24 


0° 04' 28" 
01 02 
01 58 


56.66 

84.86 

112.91 


56.50 

84 . 75 

113.00 



with the corresponding values for a straight main track (the lead 
rails circular throughout) shows that the error is insignificant in 
using the more easily computed values. 

268. Dimensions for a turnout from the INNER side of a curved 
track. (Lead rails circular throughout.) From Fig. 142 we 
have, from the triangle DFC, 




DF+FC: DF-FC :: tan i(DFC+FDC) : tan i(DFC-FDC) ; 
but i(DFC + FDO =90°-§^ 

and iiiDFC-FDC)=iF; 

/. 2R : g ::cot id : tan JF 
:-cotiFtani^; 



tanj^ = ^. 



(95) 



280 



Fiom OFCj 



RAILROAD CONSTRUCTION. 
OF'.FC ::smd:(F + d). 



sin (F + dy 
L=BF=2(R--ig)sm §<9. 



269. 

(96) 
(97) 



As in § 267, it may be readily shown that the degree of the 
turnout (d) is nearly the sum of the degree of the main track (D) 
and the degree (cV) of a turnout from a straight track when the 
frog angle is the same. The discrepancy in this case is some- 
what greater than in the other, especially when the curvature 
of the main track is sharp. If the frog angle is also large, the 
curvature of the turnout is excessively sharp. If the frog angle 
is ver}^ small, the liability to derailment is great. Turnouts to 
the inside of a curved track should therefore be avoided, unless 
the curvature of the main track is small. ^H 

269. Double turnout from a straight track. In Fig. 143 the 
frogs Fi and Fr are generally made equal. Then, if there are 




I 



Fig. 143. 

uniform curves from B^ to Fi and from B to Fr, the required 
value of Fm is obtained from 



vers iFin = 



2{r + igy 



(98) 



r being found from Eq. 78, in which n is the frog number of Fi 
or Fr. 

MFm=r tan iFm.; 

but since Um ^h cot JFm, 



MFm = 



2/15 



(99) 



§ 269. SWITCHES AND CROSSINGS. 281 

Since vers Fi=-. — rr^; 

YersiFm = iyersFi (100) 

Also, since (C,Fmy-(MFmy + (C,My, we have 

Simplifying and substituting, r=2gn'^, we have 
2g'n' + ig' = 



4g^n* 

4 



^m 



4nm'' 



2n2 + i' 

Dropping the i, which is always insignificant in comparison with 
2^^, we have 

nm=-^=nX.707(approx.) (101) 

Frogs are usually made with angles corresponding to integral 
values of n, or sometimes in "half" sizes, e.g. 6, 6^, 7, 7 J, etc. 
If No. 8J frogs are used for Fi and Fr, the exact frog number 
for F,n is 6.01. This is so nearly 6 that a No. 6 frog may be used 
without sensible inaccuracy. Numbers 7 and 10 are a less 
perfect combination. If sharp frogs must be used, 8 J and 12 
form a very good combination. 

If it becomes necessary to use other frogs because the right 
combination is unobtainable, it may be done by compounding 
the curve at the middle frog. Fi and Fr should be greater 
than ^Fm- If equal to hFm, the rails would be straight from 
the middle frog to the outer frogs. In Fig. 144, 6j^=Fi — iFm* 



Drawing the chord FiFm, 

KFjFm=Fi-id,=Fi-iFi + iF,n = i(Fi + iFm); 

KFra g 



FiFm = — =-7: — -. — rrB — , -i m \ f • • (102) 

sin KFiFm 2 sm i(Fi + iFm) 



KFi=KFmCotKFiFm==ig cot i(Fi + iFm); . (103) 



2 sin id 4 sin i(Fi + iFm) sin i{Fi -iFm) 



^^ .... (104) 



cos iFm— COS Fi 



282 



RAILKOAD CONSTRUCTION. 



§270. 



If three frogs, all different, must be used, the largest may be 
selected as Fm ; the radius of the lead rails may be found b}^ an 
inversion of Eq. 98; Fm may be located in the center of the 
tracks by Eq. 99 ; then each of the smaller frogs may be located 




Fig. 144. 

by separate applications of Eq. 112 or 103, the radius being 
determined by Eq. 104. 

270. Two turnouts on the same side. In Fig. 145, let 0^ 
bisect O2D, Then (n + ig) =i(r2 + hg); also, Ofi^-O^Fi and 

Fr=Fl. 



vers Fm = 



2g 



, 1 - , 1 , .... (105) 

r' + ig r + ig' 

BFm=(r' + ig) sin Fm (106) 

It may readily be shown that the relative values of Fr, Fi, and 
Fm are almost identical with those given in § 269; as may be 



^ 




Fig. 145. 



apparent when it is considered that the middle switch may be 
regarded simply as a curved main track, and that, as developed 



I 



§271. 



SWITCHES AND CROSSINGS. 



283 



in § 267, the dimensions of turnouts are nearly the same whether 
the main track is straight or slightly curved. 

271. Connecting curve from a straight track. The "con- 
necting curve" is the track 
lying between the frog and q ^ 
the side track where it be- 
comes parallel to the main 
track (FS in Fig. 146 or 147). 
Call d the distance between 
track centers. The angle 
FO,R=F (see Fig. 146). 
Call / the radius of the con- 
necting curve. Then 




{r'-hg) = 



vers F ' 



. (107) 

FR = {r'-^g)^mF, 



Fig. 146. 



(108) 



If it is considered that the distance FR consumes too much 
track room it may be shortened by the method indicated in 
Fig. 151. 

272. Connecting curve from a curved track to the OUTSIDE. 
When the main track is curved, the required quantities are the 




Fig. 147. 

radius r of the connecting curve from F to Sj Fig. 147, and its 
length or central angle. In the triangle CSF 

CS + CF:CS-CF:: tan i{CFS + CSF) : tan h(CFS - CSF) ; 



284 



RAILROAD CONSTRUCTION, 



§ 273. / , 



but -ACFS + CSF) =90-i(/^; and, since the triangle O^SF is 
isosceles, ^(CFS-CSF) =hF; 



.-. 2R + d:d-g::coth(lr.t8iniF 
:-cot JF:tan J^; 

••• '-i'-'i^- ■ • • 

From the triangle CO^F we may derive 

r — ig:R + ig::sm ^-sin (F + ip); 
sin (p 



(109) 



r-i9 = (R + ig); 



(110) 



sin(i^+9^) 

Also FS=2(r-ig)smi{F + (P) (Ill) 

273. Connecting curve from a curved track to the INSIDE^ 




Fig. 148. 

As above, it may readily be deduced from the triangle CFS (see 
Fig. 148) that 

(2R-d) : (c^-gr) :: cot J^: tan ^F, 

and finally that 

2n(d--g) 
2R-d 



(112) 



tan ^d) 
Similarly we may derive (as in Eq. 110) 

ir-lg)^iR-l,)^^^^ (113) 



§273. 



SWITCHES AND CROSSINGS. 



28i 



Also 



FS=2(r-ig) sin i(F-cl^) (114) 



Two other cases are possible, (a) r may increase until it 
becomes infinite (see Fig. 149), 
then F = (lf. In such a case 
we may write, by substitut- 
ing in Eq. 112, 

2R-d = 4n'(d-g). . (115) 

This equation shows the value 
of R, which renders this case 
possible with the given values 
of n, d, and g. (b) (p may be 
greater than F. As before 
(see Fig. 150) 

2R-d:d-g::cot J^itanji^; 



tan J 9^ 



2n(d 



gl 

2R-d ' 




the same as Eq. 112, but 



Fig. 149. 



r+ig=(R-i9) 



sin (ff 



sm((p-Fy 



(116) 




Fig. 150. 



Problem. To find the dimensions of a connecting curve run- 
ning to the INSIDE of a curved main track; number 9 frog, 4° 30' 
curve, d = 13', 5r = 4'8i^ 



286 



RAILROAD CONSTRUCTION. 



§274. 



Solution. 






Eq. 112. rf=13 


000 


log 2n= 1.25527 


g= 4 


708 


log id-g)= .91866 


(d-g)= 8 


292 


co-log (2jB-rf) = 6.59616 


22 = 1273.6 




log tan ^* = 8. 77009 


27? = 2547. 2 




i4' = 3°22' 14" 


272-d = 2534.2 




* = 6° 44' 28'' 


log(2i2-rf) = 3.40384 




F = 6° 21' 35" 
(*-F) = 0°22'53' 


Eq. 116. 22 = 1273.6 


/og(/2-i^) = 3.10423 


ig= 2.35 




log sin * = 9. 06960 


(R-ig) =1271.25 




co-log sin (i' - F) - 2 . 17676 


(^_ir') = 1373", log = 3. 13767 




(r + ^ff) = 22418.0. .4.35059 


4.68557 




r = 22415. 6 


log sin (*-i^) = 7.82324 




d = 0° 15' 






Eq. 114. 


2.. .0.30103 


^(*-F) = 686."5 ..2.83664 




(r-*7) = 22413.3...4.35050 


4.68557 




sin i(*-F)...7.5222T 


BinK*-i^) =7.52221 




F.^ = 149.19 2.17375 



274. Crossover between two parallel straight tracks. (See 

Fig- 151.) The turnouts 
are as usual. The cross- 
over track may be straight, 
as shown by the full lines, 
or it may be a reversed 
curve, as shown by the 
dotted lines. The reversed 
curve shortens the total 
length of track required, 
but is somewhat objection- 
able. The first method re- 
quires that both frogs must 
be equal. The second 
method permits unequal 
Fig. 151. frogs, although equal frogs 

are preferable. The length of straight crossover track is FiT. 



02 




V 








s 




1 1 
1 

1 

h 

Y 


s 






D 








FJ^ sin F^^g cos F^ =d—g] 



F^T^-^-§r-9cotF,, 



(117) 



§274. 



SWITCHES AND CROSSINGS. 



287 



The total distance along the track may be derived as follows: 

DV=2DF,-{-F,Y = 2DF, + XY-XF2; 
XY = (d-g) cotF^; XF2=g-^smF2) 

9 



DV=2DF, + {d-g) cot F, 



sinF, 



If a reversed curve with equal frogs is used, we have 

d 



vers 6 = 



also 



2r' • ■ 
DQ=2rsme 




Fig. 152. 



If the frogs are unequal, we will have (see Fig. 152) 
r2 vers d+ri vers d=d; 

d 



\ vers 6 = 



(118) 



(119) 
(120) 



(121) 



also the distance along the track 

B2N = (r,+r,) sin d (122) 

Problem, A crossover is to be placed between two parallel 
straight tracks, 12' 2'' between centers, usiag a No. 8 and a No. 9 



288 



RAILROAD CONSTRUCTION. 



§274. 



frog, and with a reversed curve between the frogs. Required 
the total distance between switch-points (the distance ^2^ i^^ 
Fig. 152). 

Solution. If straight point rails and straight frog rails are 
used, the radii, r^ and rg, taken from the middle section of Table 
III, are 527.91 and 681.16. 

vers 6 = — ; — 

ci = 12' 2" =12.16, log =1.08517 

log ('r,+r2) = 3^08245 
log vers ^ = 8.00272 



Eq. 122. 



ri = 527.91 
rz = 681.16 
r, +-2=1209.07 

Eq. 122. 



e = s° 08' 06' 



Bo.V=171.09 



log(ri+r2) =3.08245 
log sin ^ = 9.15077 
log 171.09 = 2.23322 



The length of the curve from B., = lOO{d~d) =100(8° 08' 06''-- 
8° 250 =96.65. The length of the other curve is 100(8° 08' 06' -^ 




Fig. 153. 



10° 52') =74.86. As a check, 96.65 + 74.86 = 171.51, which is 
slightly in excess of 171.09, as it should be. 



§275. 



SWITCHES AND CROSSINGS. 



289 



275. Crossover between two parallel curved tracks, (a) Using 
a straight connecting curve. This solution has limitations. If 
one frog (F^) is chosen, F2 becomes determined, being a function 
of F^. If F^ is less than some limit, depending on the width (d) 
between the parallel tracks, this solution becomes impossible. 
In Fig. 153 assume F^ as known. Then F^H=g see F^. In the 
triangle HOF2 we have 

sin HFfi : sin F^HO r.HOiFfi; 
sin F^HO = cos F, ; HF^O = 90° + Fn ; 
,\ sin HF20=cos F2. 
HO=R-\-id-ig-g sec F^; F20=R-hd-\-ig; 



^ ^ R-hid — hq — q sec F. 

cos F.^cos F. ^ — '^V-^, 

' ' R-id + ig 



(123) 



Knowing F2, 6^ is determinable from Eq. 91. Fig. 153 shows 
the case where 62 is greater than F2. Fig. 154 shows the case 
where it is less. The demonstration of Eq. 123 is applicable to 




Fig. 154. 



htjih. figtifes The relative position of the frogs F^ and F2 may 
be determined as follows, the solution being applicable to both. 
Figs. 153 and 154: 



Then 



Gi^i=2(i^ + J^-k)sinKi^i~i^2) (124) 



Since F2 comes out any angle, its value will not be in general 
that of an even frog number, and it will therefore need to be 
made to order. 



290 



RAILROAD CONSTRUCTION. 



§275. 



(b) Continuing the switch-rail curves until they meet as a 
reversed curve. In this case F^ and F2 may be chosen at pleasure 
(within limitations), and they will of course be of regular sizes 
and equal or unequal as desired. F^ and Fg being known, 0^ 
and 62 are computed by Eq. 95 and 91. In the triangle 00^2 
(see Fig. 155) 

2{S-002){S-00,) 
vers ^ (002)(00i) 

in which S = ^{00^ + OO2 + Ofli) ) 

but 00i=i^ + ic?-ri, 

002=R-hd-r2, 

.-. S = h{2R-\-2r2)=R+r2] 
S-002=R + r2-R + id-r2 = id; 
S-00^=R + r2-R-id + r^=r,-hr2-id; 




Fig. 155. 



'^^^^'^~ {R-hd^r2){R-\-hd-r,)' ' ' 
m 00 A =sm ^^^^=sm <p ^^^^^ ; . 

020J) = (p-\-Ofi20] 

NF2 = 2{R-^d + lg) sin ^{^-6^-62), 



sm 



(125) 

(126) 

(127) 
(128) 



§ 275. SWITCHES AND CROSSINGS. 291 

Although the above method introduces a reversed curve, yet 
it uses up less track than the first method and permits the use of 
ordinary frogs rather than those having some special angle wliich 
must be made to order. 

Problem. Required the dimensions of a crossover on a 4° 30' 
curve when the distance between track centers is 13 feet. The 
I frog for the outer main track {F^ in Fig. 155) is No. 9 ; F2 is No. 7. 
Then 7^ = 1273.6; R^, for the inner main track, =1280.1; D^== 
4° 29'; ^2 = 1267.1; Z)2=4°31'; ri=radius for {d,-\-D,y curve = 
radius for (8° 25' + 4° 29') curve =445.09 ; r^ =radius for {d^-D^Y 
curve =radius for (14° 27' -4° 31') curve =577.53. (See §§ 267- 
268.) 

Eq. 125. rf=13; log=l. 11394 

ri+r2-ic?=1016.12-; log = 3.00694 

72- ^d + r2= 1844.63; log = 3 . 26586 ; co-log = 6. 734 14 

i2 + id-ri= 835.01; log = 2. 92169; co-log = 7^07831 

<\r = 7° 30^ 35'^ log vers ^ = 7 . 93334 

Eq. 126. log sin * = 9 . 1 1626 

log (72 + i(i-ri) = 2. 92169 

r^ + rs =1022.62; log = 3. 00971; co-log =^ 6 . 99028 

00-201-6° or 34^^ sin OOgO] = 9 . 02823 

Eq. 127. 020il> = 7° 30' 35" -H 6° 07' 34'' = 13° 38' 09'' 



72 20 
Lead from sT\atch point No. 1 up to F^=72.20]d^ = ~—^d'j 

where d' corresponds to the radius {R-\-\d — ^g) or 1277.75; 
(i'=4° 29'; (9i=3° 14'. 

r» -I £? (T 

Lead from switch point No. 2 up to i^2 = 61.65; 62= d" , 

where d" corresponds to the radius {R — \d-\-\g) or 1269.45; 
d"=4° 31'; <92^2° 47'. 

Eq. 128. 2; log =0.30103 

7^-1(^ + 1^ = 1269.45: log = 3. 10361 

§(^-6>i-6^2)=0°44'48"; log sin =8. 11497 

A^F2=33.08. , log33.08 = 1.5196r 

13° 38' 09" 
Length of curve with radius r^ = 100 -.00 r./ — = 105 . 70 ; 



6° 07' 34 



// 



" " " " " ''2 = 100-^0-5^,— = 61.67; 

Total length of curve between swatch points = 1 67 . 37. 
As a check, the sum of the two leads and .Yi^2 equals (72.20 + 



292 



RAILROAD CONSTRUCTION. 



§276. 



61 . 65 + 33 . 08) = 166 . 93, wliich is a little less than the length of 
the curve, as it should be. 

Note that the point of reversed curve is placed .02'(=i'0 
beyond the frog point i^2- I^ the computations had apparently 
indicated the point of reversed curve coming between the frog 
point and the switch point, it would have shown the impracti- 
cability of the combination of No 7 and No. 9 frogs with this 
particular degree of curve, gauge of track, and distance between 
track centers. If both frogs were made No 9 the total length 
of track between switch points w^ou3d be increased to over 188 
feet and the point of reversed curve would be nearly at the middle 
point. This shows that the frog numbers should be nearly equal, 
but also shows that there is some choice *' within limitations /' 

276. Practical rules for switch-laying. A consideration of 
the previous sections will show that the formulae are compara- 
tively simple when the lead rails are assumed as circular; that 
they become complicated, even for turnouts from a straight 
main traek, v/hen the effect of straight frog and point rails is 
allowed for, and that they become hopelessly complicated when 
allowing for this effect on turnouts from a curved main track. 
It is also sho\Mi (§ 267) that the length of the lead is practically 



/' VMDN=a: 

^ll — i ry 



I 
I 

I 




Fig. 140. 



the same whether the main track is straight or is curved with 
such curves as are commonly used, and that the degree of curve 
of the lead rails from a curved main track may be found with 



§276. 



SWITCHES AND CROSSINGS. 



293 



close approximation by mere addition or subtraction From 
this it may be assumed that if the length of lead (/.) and the 
radius of the lead rails (r) are computed from Eq 87 and 90 for 
various frog angles, the same leads may be used for curved main 
track* also, that the degree of curve of the lead rails may be 
found by addition or subtraction^ as indicated in § 267, and that 
the approximations involved will not be of practical detriment 
In accordance with this plan Table III has been computed from 
Eq. 87, 88, and 90 The leads there given may be used for all 
main tracks, straight or curved. The table gives the degree of 
curve of the lead rails for straight main track; for a turnout to 
the inside J add the degree of curve of the main track ; for a turn- 
out to the outside, subtract it- 

If the position of the switch-block is definitely determined, 
then the rails must be cut accordingly; but when some freedom 
is allowable (which never need exceed 15 feet and may require 
but a few inches), one rail-cutting may be avoided. Mark on 
the rails at B, F, and D (see Fig. 140) ; measure off 
the length of the switch-rails DN\ offset \g-\-k from 
N for the point S. The point H may be located 
(temporarily) by measuring along the rail a distance 
FH{ =/) and then swinging out a distance ol 
f^n (n being the frog number). HT = ^g and is 
measured at right angles to FH, Points for track 
centers between S and T may be laid off by a transit 
or by the use of a string and tape. Substituting in 
Eq. 31 the value of R and of chord ( =ST), we may 
compute x(=db). Locate the middle point d 
and the quarter points a'' and c'\ Then a' 'a and 
c'^c each equal three-fourths of db. Theoretically 
this gives a parabola rather than a circle, but the 
difference for all practical cases is too small for measurement. 

Example. Given a main track on a 4° curve ; a turnout to 
the outside, using a number 9 frog; gauge 4' 8i''; / = 3'.37; 
/c=5r'; DN = 15' 0'' and a = l° 50'. Then for a straight track 
r would equal 631.16 [rf = 9° 05' J. For this curved track d will 
be nearly (9° 05'~4°)=5° 05', or r will be 1131.2. L for the 
straight track would be 72.20; but since the lead is slightly 
increased (see § 267) w^hen the turnout is on the outside of a 
curve, L may here be called 72.5. i^//=/=3'.37; /H-r? = 
3.37-^9=0',375=4".5. H, T, and S mav be located as de- 




Fig. i'SO. 



294 



RAILKOAD CONSTRUCTION. 



§277. 



scribed above. ST may be measured on the ground, or it may 
be computed from Eq, 88, giving the value of 53.80 feet for 
straight track. Since it is shghtly more for a turnout to the 
outside of a curve, it may be called 54.0. Then x=db = 

= . 322 foot, and aa'' and cc'' = . 24 foot. 

Wl 

CROSSINGS. " ' 



8X1131.2 



277. Two straight tracks. When two straight tracks cross 
each other, four frogs are necessary, the angles of two of them 




being supplementary to the angles of the other. Since such 
crossings are sometimes operated at high speeds, they should be 



§278. 



SWITCHES AND CROSSINGS. 



295 



Structurally the 



very strongly constructed, and the angles should preferably be 
90° or as near that as possible. The frogs will not in general 
be ''stock'' frogs of an even number, especially if the angles are 
large, but must be made to order with the required angles as 
measured. In Fig. 157 are shown the details of such a crossing. 
Note the fillers, bolts, and guard-rails. 

278. One straight and one curved track, 
crossing is about the same as above, 
but the frog angles are all unequal. 
In Fig. 158, R is kno^\Ti, and the 
angle M, made by the center lines 
of the tracks at their point of inter- 
section, is also knoAvn. 

M = NCM. NC =Rcos M. 

{R - hg) cos i^i = NC + ^g ; 

R cos Al + ig 



cosF^ = 



Similarly 



R-hg 



cos 1^ 



, _ R cos M + ^g 

'~ R + ig ^' 



cos Fo = 



RcosM — hg 



cos F^ = 



R + lg ' 

R cos M — hg 



R-hg 



(129) 




Fig. 158. 



(130) 



F,F, = (R + hg) sin F, -{R- ^g) sin F, ; 
HF, = {R- l,g) (sin F, - sin F,) . 

279. Two curved tracks. The four frogs are unequal, and 
the angle of each must be computed. The radii R^ and R2 are 
known; also<the angle M. r^, r^^ rj, and r4 are therefore known 
by adding or subtracting \g, but the lines are so indicated for 
brevity. Call the angle MC^C^^C^, the angle MC2C^ = C2, and 
the line Cfi^^c. Then 

«C'i + C2)=90°-iM 
and 



tanKCi-C2)=cotp/ 
Ci and C2 then become known and 

sm (7i 



R2z-_Ri 

R2-\-Ri 



(131) 



(132) 



296 hailroad coNSTHtrcrioN. § 279. * 

In the triangle i^ACz, call iC^+rj+rJ =Si; S2 = i{c + r2+r^); 




4 



Fig. 159. 



S3 = i(c + ri-\-r.^); and §4 = 4(^ + ^2 + ^3)- Then, by formula 29, 
Table XXX, 

2(si-ri)(si-r4) 



vers jPi = 



r,r 



v i 



Similarly 



vers F2 = -^^ ^^^^ , 

^2^ 



vers Fo = 



2(-^3-n)(^^3-n) 



r,r 



1'3 



vers F, = 



2(54-r2)(54-r3) 



rnV, 



(133) 



i 



sin CiCiF^ = sin i^^— ; 

sinCA^2 = sini^2-; 
c 

F^C.,F^=Cfi,F^-Cfi^F.„ (134) 



sin i^iCCz = sin i^i-^; 



sin7^2<^iC2=sinF2-, 



I 



.-. F,C,F,=F,Cfi,-F,C,C^; (135) 

from which the chords F^F-^ and F^Fi are readih' computed. M-^ 



I 



§279. 



SWITCHES AND CROSSINGS. 



297 



F1F2 and F2F^ are nearly equal. When the tracks are straight 
and the gauges equal, the quadrilateral is equilateral. 

PrGhlem. Required the frog angles and dimensions for a cross- 
ing of two curves (Z)i=4°; D2 = S°) when the angle of their tan- 
gents at the point of intersection =62° 28' (the angle AI in 
Fig. 159). 



Solution 



Eq. 131. 



i^i = 1432.7; 7?2 = 1910.1; 
Ti =i?2 + Js^ = 1910.1+2.35 = 1912.45; • 
^2 =i^,-ig = 1910.1-2.35 = 19C7.75; 
r, =i^i + i9' = 1432. 7 + 2. 35 = 1435. 05; 
U =i?i-i^ = l-132.7-2.35 = 1430.35. 

log cot pf =0.21723 
i^2-7?i=477.4; log =2.67888 

i^2 + i?i=3342.8; log=3. 52411; co-log = 6 .47589 

KC'i-Co) =13° 15' or'; tan 13° 15' 07"=9. 37200 
K^i + Cs) =58° 46' [-K^i + Q =90°- JM] 



58° 46' 

(?i=72°01'07" 
(7^ = 45° 30' 53" 



Eq. 132. 



c==C,C2 = 1780.7: 

Eq. 133. 
c = 17S0.7 
ri = 1912.45 
r4 



1430 ■ 35 

_2|5123.50 

Si = 2561 .75 

si-ri= 649.30 

5i-r4=1131.40 



log sin Ci 



c = 1780.7 
r2= 1907.75 
7-4=1430.35 



_2|5118.80 

S2 = 2559.40 

S2-r2= 651.65 

-r4 = 1129.05 



logi?2=3.28105 

log sin M = 9. 94779 

9.97825; co-log = 0.02175 

log(7iC2=3.25059 



c=1780.7 

r2= 1907.75 

r3 =1435.05 

_2|5123.50 

S4 = 2561.75 

S4-r2= 654.00 

S4-r3= 3126.70 



c = 


1780 


7 


ri = 


1912 


45 


rz = 


1435 


05 


2 


5128 


20 


§3 = 


2564 


.10 


-r,= 


651 


.65 


-r3 = 


1129 


.05 



ri = 1912.45; 
r4=1430.35; 
Fi=62° 25' 31 



r2= 1907.75; 
r4=1430.35; 

F., = 62- 33' 55' 



log = 3.28159; 
log = 3. 15544; 



log = 3. 28052 ; 
log = 3. 155445 



log 2 = 0.30103 

(si-n); log 649.30 = 2.81244 

{si-n)\ log 1131.40 = 3.05361 

co-log = 6. 7 1841 

co-log = 6.84456 

log vers 62° 25 ' 31" = 9 .73006 

log 2 = 0.30103 

(s2-r2); log 651.65 = 2.81401 

(s2-r4); log 1129.05 = 3.05271 

co-log = 6. 7 1948 

co-log = 6_84456 

log vers 62° 33' 55" = 9. 73180 



298 



RAILROAD CONSTRUCTION. 



§ 279. 



ri = 1912.45; log = 3.28159; 
rg = 1435 . 05 ; log = 3 . 15686 ; 



r2= 1907.75; log = 3. 28052; 
rs = 1435 . 05 ; log = 3 . 15686 ; 



log 2 = 0.30103 

(s3-ri); log 651.65 = 2.81401 

(s3-^3); log 1129.05 = 3.05271 

co-log = 6. 7 1841 

co-log = 6. 84 313 

log vers 62° 21' 57" = 9. 72930 



\L 



62° 30' 14' 



log 2 = 0.30103 

(s4-r2); log 654.00 = 2.81558 

(«4-r3); log 1126.70 = 3.05181 

co-log = 6. 7 194 8 

co-log = 6.84313 

log vers 62° 30' 14" = 9. 73103 



As a check, the mean of the frog angles = 62° 27' 54 ', which is within 6" of 
the value of M, 



Eq. 134. 



CiC2i^4 = 45° 37' 51"; 



log c = 3. 25059; 



CiC2F2 = 45° 28' 17"', 

^2^2^4 = 45° 37' 51" -45* 28' 17" = 0° 09' 34' 



K0°09' 34") = 0°04'47' 



F?F.- = 5.309 ; 
p:q. 135. 

FiCiC2=72° 10' 22"S 



l?»^CiC2=7P57'38"] 

i?^,CiF2 = 72*> IC 22"-71*» 57' 38" = 0° 12' 44". 



log sin 2^4 = 9.94794 

log r3 = 3. 15686 
co-log c = 6. 74940 

sin 01^2^4 = 9.85421 

logsini?'2 = 9.94818 

log r4 = 3. 15544 

co-log c = 6. 74940 

sin CiC2F2 = 9.85303 



log 2 = 0.30103 

log r2 = 3. 28052 

log sin =^4- 685^^ 
iogsin-^^2^^^788 

log Jfi^2i^4 = 0.72500 



K0° 12' 44") = 0°06' 22" 



F,F, = 5.298- 



sin Fi = 9. 94763 

log ri = 3. 28159 

co-log c = 6. 74940 

sin J^iCiC2 = 9. 97863 

sin ^2 = 9. 94818 

log r2 = 3. 28052 

co-log c = 6. 74940 

sin jP2CiC2 = 9.97811 



log 2 = 0.30103 
log r4 = 3. 15544 

log sin =/'4- 68557 
V 2. 58206 

logFiF2 = 0.72411 



As a check, F2F^ and F^F2 are very nearly equal, as they should 



be. 



CHAPTER XII. 

MISCELLANEOUS STRUCTURES AND BUILDINGS. 

WATER-STATIONS AND WATER-SUPPLY. 

280. Location. The water-tank on the tender of a locomo- 
tive has a capacity of from 2500 to 5000 gallons — sometimes less, 
rarely very much more. The consumption of water is very vari- 
able, and mU correspond very closely with the work done by 
the engine. On a long down grade it is very small; on a ruling 
grade going up it may amount to 150 gallons per mile in ex' 
ceptional cases, although 60 to 100 gallons would be a more usual 
figure. Nominally a locomotive could run 40 miles or more on 
one tankful, but it would be impracticable to separate the water- 
stations by such an interval. On roads of the smallest traffic, 
15 to 20 miles should be the maximum interval between stations; 
10 miles is a more common interval on heavy traffic-roads. But 
these intervals are varied according to circumstances. In the 
early history of some of the Pacific railroads it was necessary to 
attach one or more tank-cars to each train in order to maintain 
the supply for the engine over stretches of 100 miles and over 
where there was no water. Since then water-stations have been 
obtained at great expense by boring artesian wells. The indi- 
vidual locations depend largely on the facility Tvdth which a suffi- 
cient supply of suitable water may be obtained. Streams inter- 
secting the railroad are sometimes utilized, but if such a stream 
passes through a limestone region the water is apt to be too hard 
for use in the boilers. More frequently wells are dug or bored. 
WTien the local supply at some determined point is unsuitable, 
and yet it is necessary to locate a water-station there, it may 
be found justifiable to pipe the water several miles. The con- 
struction of municipal water-works at suitable places along the 
line has led to the frequent utilization of such supplies. In such 
cases the railroad is generally the largest single consumer and 
obtains the most favorable rates. When possible, water-stations 
aie located at regular stopping points and at division termini. 

^99 



300 RAILROAD CONSTRUCTION. § 281. 

281. Required qualities of water. Chemically pure water is 
unknown except as a laboratory product. The water supplied 
by wells, springs, etc., is alwa3^s more or less charged with cal- 
cium and magnesium carbonates and sulphates, as well as other 
impurities. The evaporation of water in a boiler precipitates 
these impurities to the lower surfaces of the boiler, where they 
sometimes become incrusted and are difficult to remove. The 
protection of the iron or steel of a boiler from the fierce heat of 
the fire depends on the presence of water on the other side of the 
surface, which will absorb the heat and prevent the metal from 
assuming an excessively high temperature. If the w^ater side 
of the metal becomes covered or incrusted with a deposit 
of chemicals, the conduction of heat to the water is much less 
free, the metal will become more heated and its deterioration or 
destruction will be much more rapid. An especially common 
effect is the production of leaks around the joints between tubes 
and tube-sheets and the joints in the boiler-plates. Such in- 
jury can only be prevented by the application of one (or both) 
of two general methods — (a) the frequent cleaning of the boilers 
and (h) the chemical purification of the water before its intro- 
duction into the boiler. Although ^'manholes" and ''hand- 
holes'' are made in boilers, it is physically impossible to clean 
out every corner of the inside of a boiler where deposits will form 
and where they are especially objectionable — on the tube-sheets. 
Such a cleaning is troublesome and expensive. 

Chemical purification is generally accomplished by treating 
the water before it enters the boiler. The reagents chiefly em- 
ployed are quicklime and sodium carbonate. Lime precipi- 
tates the bicarbonate of lime and magnesia. Sodium carbo- 
nate gives, by double decomposition in the presence of sulphate 
of lime, carbonate of lime, which precipitates, and soluble sul- 
phate of soda, which is non-incrustant. When this is done in a 
purifying tank, the purified water is drawn off from the top of 
the tank and supplied pure to the engines. The precipitant s are 
drawn off from the settling-basin at the bottom of the tank. 
This purification, w^hich makes no pretense of being chemically 
perfect, may be accomplished for a few cents per 1000 gallons. 
It is used much more extensively in Europe than in this country, 
the Southern Pacific being the only railroad w^hich has employed 
such methods on a large scale. Reliance is frequently placed 
on the employment of a '^ non-incrustant " which is introduced 



§ 282. MISCELLANEOUS STRUCTURES AND BUILDINGS 301 

directly into the boiler. When no incrustation takes place 
the accumulation of precipitant and mud in the bottom of the 
boiler may be largely removed b}^ mere *' blowing cff" or by 
washing out with a hose. 

American practice may therefore be summarized as follows: 
(a) Employing as pure water as possible; (6) cleaning out boil- 
ers by ^' blowing off " or by washing out with a hose or by physi- 
cal scraping at more or less frequent intervals or w^hen other 
repairs are being made; (c) the occasional employment of non- 
incrustants; (d) the occasional chemical treatment of water be- 
fore it enters the tender-tank. 

282. Tanks. Whatever the source^ the water must be led 
or pumped into tanks which are supported on frames so that the 
bottoms of the tanks are about 12 feet above the rails. Wooden 

TABLE XIV. CAPACITY OF CYLINDRICAL Vv^lTER-TANKS IN 

UNITED STATES STANDARD GALLONS OF 231 CUBIC INCHES. 



Height 






Diameter of tank in feet. 






in 


















feet. 


10 


12 


14 


16 


18 


20 


22 


24 


6 


3525 


5076 


6909 


9024 


11421 


14101 


17062 


20305 


7 


4113 


5922 


8061 


10528 


13325 


16451 


19905 


23689 


8 


4700 


6768 


9212 


12032 


15229 


18801 


22749 


27073 


9 


5288 


7614 


10364 


13536 


17132 


21151 


25592 


30457 


10 


5875 


8460 


11515 


15041 


19036 


23501 


28436 
31280 


33841 


11 


6463 


9306 


12667 


16545 


20939 


25851 


37225 


12 


7050 


10152 


13819 


18049 


22843 


2320 1 


34123 


40609 


13 


7638 


10998 


14970 


19553 


24746 


30551 


36967 


43994 


14 


8225 


11844 


16122 


21057 


26650 


32901 


39810 


47378 


15 


8813 


12690 


17273 


22561 


28554 


35251 


42654 


50762 


16 


9400 


13536 


18425 


24065 


30457 


37601 


45498 


54146 


17 


9988 


14383 


19576 


25569 


32361 


39951 


48341 


57530 


18 


10575 


15229 


20728 


27073 


34264 


42301 


51185 


60914 


19 


11163 


16075 


21879 


28577 


36168 


44652 


54028 


6^298 


20 


11750 


16921 


23031. 


30081 


38071 


47002 


56872 


67682 


21 


12338 


17767 


24182 


31585 


39975 


49352 


59716 


71067 


22 


12925 


18613 


25334 


33089 


41879 


51702 


62559 


74451 


23 


13513 


19459 


264S5 


34593 


43782 


54052 


65403 


77835 


24 


14101 


20305 


27637 


36097 


45686 


56402 


68246 


81219 


25 


14688 


21151 


28789 


37601 


47589 


58752 


71090 


84603 



tanks having a diameter of 24 feet, 16 feet high, and with a 
capacity of over 50,000 gallons are frequently employed. Iron 
or steel tanks are also used. 

In Table XIV is shown the capacity of cylindrical water-tanks 
in United States standard gallons of 231 cubic inches. Froni 



302 



RAILROAD CONSTRUCTION. 



§282. 



this table the dimensions of a tank of any desired capacity 
may readily be found. Two or more tanks are sometimes used 
rather than construct one of excessive size. The smaller sizes 
shown in the table are of course too small for ordinary use, 
but that part of the table was filled out for its possible con- 
venience otherwise. On single-track roads where all engines 
use one track the tank may be placed 8' 5" from the track 
center; this gives sufficient clearance and yet permits the use 
of a single swinging pipe which will reach from the bottom 
of the tank to the tender manhole. In Fig. 160 is illustrated 




Fig. 160. — Water-tank. 

one form of wooden tank. They are preferably manufactured 
by those who make a special business of it and who by the use 
of special machinery can insure tight joints^ When it is incon- 
venient to place the tank near the track, or when there is a 
double track, a ^stand-pipe" becomes necessary. See § 285. 
One of the most difficult and troublesome problems is to prevent 
freezing, particularly in the valves and pipes Not only are the 
pipes carefully covered but fires must be maintained during cold 
weather. When the pumping is accomplished by means of a 
steam-pump, supplied from a steam-boiler in the pump -house 
under the tank, coils of steam-pipe may be employed to heat the 
water or to heat the pipes Partial protection may be obtained 
by means of a double roof and double bottom, the spaces being 
filled with sawdust or some other non-conductor of heat. 



I 



§ 283. MISCELLANEOUS STRUCTURES AND BUILDINGS. 303 

283. Pumping. The pumping is done most reliably with 
steam-pumps or gas-engines, although hot-air engines/windmills, 
and even man-power are occasionally employed. Economy of 
operation requires that the water-stations shall be so located 
that each tank shall be used regularly and that each pump shall 
be regularly operated for maintaining the w^ater-supply. On 
the other hand, the pump should not be required to regularly 
work at night to maintain the supply and should have an excess 
capacity of sa}^ 25%. When a tank is but little used, it will still 
require the labor of an attendant, and his time will be largely 
wasted unless he can be utilized for other labor about the station. 
In recent years gasoline has been extensively employed as a fuel 
for the pumping-engines. The chief advantages of its use lies in 
the extreme simplicity of the mechanism and the very slight 
attention it requires, which permits their being operated by 
station-agents and others, who are paid $10 per month extra, 
instead of paying a regular pumper vS35 per month. '^ Screen- 
ings, " ^^ slack coal,'' etc., are used as fuel for steam-pumps and 
may frequently be delivered at the pump-house at a cost not 
exceeding 30 cents per ton, but even at that price the cost of 
pumping per thousand gallons, although dependent on the hori- 
zontal and vertical distances to the source of supply and to 
the tank, will generally run at 2 cents to 6 cents per 1000 gallons. 
In many cases where steam plants have been replaced by gasohne 
plants, the cost of pumping per 1000 gallons has been reduced 
to one third or even one fourth of the cost of steam pumping. 
Of course the cost, using windmills, is reduced to the mere 
maintenance of the machinery, but the unreliability of wind as 
a motive powder and the possibility of its failure to supply water 
when it is imperatively needed has made this form of motive 
power unpopular. (See report to Ninth Annual Convention 
of the Association of Railway Superintendents of Bridges and 
Buildings, Oct. 1899.) 

284. Track tanks. These are chiefly required as one of the 
means of avoiding delays during fast-train service. A trough, 
made of steel plate, is placed between the rails on a stretch of 
perfectly level track. A scoop on the end of a pipe ts loT\'ered 
from under the tender into the tank while the train is in motion. 
The rapid motion scoops up the water, which then flows 'nto the 
tender tank. The following brief description of an 'nstallation 
on the Baltimore & Ohio Railroad between Baltimore and 



304 RAILROAD CONSTRUCTION. § 2S5. 

Philadelphia will answer as a general description of the 
method. The trough is made of -^^" steel plate, 19'' wide; 6'' 
deep, and has a length of 1200 feet. There is riveted on each 
side a line of 1 J'' X2'' X \'^ angle bars. These angle bars rest on 
the ties. Ordinary track spikes hold these angle bars to the 
ties, but permit expansion as with rails. The tanks are firmly 
anchored at the center, the ends being free to expand or con- 
tract. The plates are 15 feet long and are riveted with -{^'^ 
rivets, 20 rivets per joint. At each end is an inclined plane 
13' S" long. If the fireman should neglect to raise the scoop 
before the end of the tank is reached, the inclined plane will 
raise it automatically and a catch will hold it raised. Water 
is supplied to the tanks by a No. 9 Blake pump having a 
capacity of 260 gallons per minute. During cold weather, 
freezing is prevented by injecting into the side of the tanks, 
at intervals of 45 feet, jets of steam, w^hich come through 
J" holes. Two boilers of 80 and 95 H.P. are required for pump- 
ing and to keep the water from freezing. During warm 
weather an upright 25 H.P. boiler suffices for the pumping. 
The cost of installation was about $10,000 to $11,000, the cost of 
maintenance being about $132.50 per month. 

285. Stand-pipes. These are usually manufactured by those 
who make a specialty of such track accessories, and who can 
ordinarily be trusted to furnish a correctly designed article. In 
Fig. 161 is shown a form manufactured by the Sheffield Car Co. 
Attention is called to the position of the valve and to the device 
for holding the arm parallel to the track when not in use so that 
it will not be struck by a passing train. When a stand pipe is 
located between parallel tracks, the strict requirements of clear- 
ance demand that the tracks shall be bowed outward slightl}^ 
If the tracks were originally straight, they may be shoved over by 
the trackmen, the shifting gradually running out at about 100 
feet each side of the stand-pipe. If the tracks were originally 
curved, a slight change in radius w^ill suffice to give the necessary 
extra distance between the tracks. 

BUILDINGS. 

286. Station platforms. These are most commonly made of 
planks at minor stations. Concrete is used in better-class work, 
also paving brick. An estimate of the cost of a platform of paving 
brick laid at Topeka, Kan., was $4.89 per 100 square feet when 



§ 2S6. MISCELLANEOUS STRUCTURES AND BUILDINGS. 305 

laid flat and $7.24 per 100 square feet when laid on edge. The 
curbing cost 36 cents per linear foot. Cinders, curbed by timbers 






>>-' 




Fig. 161. — Stand-pipe. 
or stone, bound by iron rods, make a cheap and fairly durable 
platform, but in wet weather the cinders will be tracked into 



306 RAILROAD CONSTRUCTION. § 287. 

the stations and cars. Three inches of crushed stone on a 
cinder foundation is considered to be still better, after it is once 
thoroughly packed, than a cinder surface. 

Elevation. — The elevation of the platform with respect to 
the rail has long been a fruitful source of discussion. Some roads 
make the platforms on a level with the top of the rail, others 
3'' above, others still higher. As a matter of convenience to 
the passengers, the majority find it easier to enter the car from 
a high platform, but experience proves that accidents are more 
numerous with the higher platforms, unless steps are discarded 
altogether and the cars are entered from level platforms, as is 
done on elevated roads. As a railroad must generally pay 
damages to the stumbling passenger, they prefer to build the 
lower platform. Convenience requires that the rise from the 
platform to the lowest step should not be greater than the rise 
of the car steps. This rise is variable, but with the figures usually 
employed the application of the rule will make the platform 
5'' to 15'' above the rail. 

Position with respect to tracks. — Low platforms are gen- 
erally built to the ends of the ties, or, if at the level of the top 
of the rail, are built to the rail head. Car steps usually 
extend 4' 6" from the track center and are 14'' to 24" above the 
rail. The platform must have plenty of clearance, and when 
the platform is high its edge is generally required to be 5' 6" 
from the track center. 

287. Minor stations. For a complete discussion of the design 
of stations of all kinds, including the details, the student is re- 
ferred to ^'Buildings and Structures of American Railroads,'^ 
by Walter G. Berg, now Chief Engineer of the Lehigh Valley 
Railroad. The subject is too large for adequate discussion here, 
but a few fundamental principles will be referred to. 

Rooms required. An office and waiting-room is the mini- 
mum. A baggage-room, toilet-rooms, and express office are 
successively added as the business increases. In the Southern 
States a separate waiting-room for colored people is generally 
provided. It used to be common to have separate waiting-rooms 
for men and women. Experience proved that the men's wait- 
ing-room became a lounging place and smoking-room for loafers, 
and now large single waiting-rooms are more common even in 
the more pretentious designs, smoking being excluded. The 
office usually has a bay window, so that a more extended view 



§ 288. MISCELLANEOUS STRUCTURES AND BUILDINGS. 307 

of the track is obtainable. The Avomen's toilet-room is entered 
from the waiting-room. The men's toilet-room, although built 
immediately adjoining the other in order to simphfy the plumb- 
ing, is entered from outdoors. Old-fashioned designs built the 
station as a residence for the station-agent; later designs have 
very generally abandoned this idea. ''Combination" stations 
(passenger and freight) are frequently built for small local 
stations, but their use seems to be decreasing and there is now a 
tendency to handle the freight business in a separate building. 

288. Section-houses. These are houses built along the right- 
of-way by the railroad company as residences for the trackmen. 
The liability of a wreck or washout at any time and at any part 
of the road, as well as the convenience of these houses for ordinary 
track labor, makes it all but essential that the trackmen should 
live on the right-of-v\'ay of the road, so that they may be easily 
called on for emergency service at any time of day or night. 
This is especially true when the road passes through a thinly 
settled section, where it would be difficult if not impossible to 
obtain suitable boarding-places. It is in no sense an extrava- 
gance for a railroad to build such houses. Even from the direct 
financial standpoint the expense is compensated by the corre- 
sponding reduction in wages, which are thus paid partly in free 
house rent. And the value of having men on hand for emergen- 
cies wdll often repay the cost in a single night. Where the coun- 
tr}^ is thickl}'- settled the need for such houses is not so great, and 
railroads will utilize or perhaps build any sort of suitable house, 
but on Southern or Western roads, where the need for such 
houses is greater, standard plans have been studied with great 
care, so as to obtain a maximum of durability, usefulness, com- 
fort, and economy of construction. (See Berg's Buildings, etc., 
noted above.) On Northwestern roads, protection against cold 
and rain or snow is the chief characteristic; on Southern roads 
good ventilation and durability must be chiefly considered. 
Such houses may be divided into two general classes — (a) those 
which are intended for trackmen only and which may be built 
with great simplicity, the only essential requirements being a 
living-room and a dormitory, and (b) those which are intended 
for families, the houses being then distinguished as ''dwelling- 
houses for employees. 

289. Engine-houses. Small engine-houses are usually built 
rectangular in plan. Their minimum length should be some- 



308 



RAILROAD CONSTRUCTION, 



§289. 



what greater than that of the longest engine on the road. They 
ma}^ be built to accommodate two engines on one track, but 
then they should be arranged to be entered at either end, so that 
neither engine must wait for the other. In width there may be 
as many tracks as desired, but if the demand for stalls is large, 
it will probably be preferable to build a '' roundhouse/^ Rect- 
angular engine-houses are usually entered by a series of parallel 
tracks switching off from one or more main tracks, no turn-table 
being necessary. If a turn-table is placed outside (because one 



8x12 




/ , ^Concrete 
-80 — 



Fig. 162. — Engine-house. 



is needed at that part of the road) enough track should be allowed 
between the house and the turn-table so that engines may be 
quickly removed from the engine-house in case of fire without 
depending on the tTirn-table to get them out of danger. 

Roundhouses. The plan of these is generally poh^gonal 
rather than circular. The straight walls are easier to build; the 
construction is more simple, and the general purpose is equally 
well served. They may be built as a part of a circle or a com- 
plete circle, a passageway being allowed, so that there are two 
entrances instead of one. When space is very limited a round- 
house w^ith turn-table will accommodate more engines in pro- 
portion to the space required (including the approaches) than a 
rectangular house. , The enlarged space on the outer side of each 
segment of a roundhouse furnishes the extra space which is needed 
for the minor repairs which are usually made in a roundhouse. 
One disadvantage is that supervision is not quite so easy or effec- 



§ 290. MISCELLANEOUS STRtJCTtrRES AND BUILDINGS. 309 

tive as in rectangular houses. Of course such houses are \ised 
not only for storing and cleaning engines, but also for minor 
repairs which do not require the engine to be sent to the shops 
for a general overhauling. 

Construction. The outer walls are usually of brick. The 
inner walls consist almost entirely of doors and the piers between 
them, although there is usually a low wall from the top of the 
door frames to the roof line, which usuall}^ slopes outward so as 
to turn rain-water away from, the central space. 

Roofs. Many roofs have been built of slate with iron truss 
framing, with the idea of maximum durability. The slate is good, 
but experience shows that the iron framing deteriorates very 
rapidly from the action of the gases of combustion of the engines 
which must be ^' fired" in the houses before starting. Ptoof 
frames are therefore preferably made of wood. 

Floors. These are variously constructed of cinders, wood, 
brick, and concrete. Brick has been found to be the best ma- 
terial. Anything short of brick is a poor economy; concrete is 
very good if properly done but is somew^hat needlessly expensive. 

Ventilation. This is a troublesome and expensive matter. 
The general plan is to have '' smoke-jacks" which drop down 
over the stack of each engine as it reaches its precise place in its 
stall and which will carry away all smoke and gas. Such a 
movable stack is most easily constructed of thin metal — say 
galvanized iron — but these will be corroded by the gases of 
combustion in two or three years. Vitrified pipe, cast iron, 
expanded metal and cement, and even plain wood painted with 
^'fireproof" paint, have been variously tried, but all methods 
have their unsatisfactory features. (For an extended discus- 
sion of roundhouse floors and ventilation see the Proc. Assoc, 
of Railway Supts. of Bridges and Buildings for 1898, pp. 112-135.) 

SNOW STRUCTURES. 

290. Snow-fences. Snow structures are of two distinct 
kinds — fences and sheds. A snow-fence implies drifting snow — 
snow carried by wind — and aims to cause all drifting snow to be 
deposited away from the track. Some designs actually succeed 
in making the wind an agent for clearing snow from the track 
where it has naturally fallen. A snow-fence is placed at right 
angles to the prevailing direction of the wind and 50 to 100 feet 
away from the tracks. When the road line is at right angles to 



310 RAILROAD CONSTRUCTION. § 291. 

the prevailing wind, the right-of-way fence may be built as a 
snow-fence — high and with tight boarding. Hedges have some- 
times been planted to serve this purpose. When the prevailing 
wind is oblique, the snow fences must be built in sections w^here 
they will serve the best purpose. The fences act as wind break- 
ers, suddenl}^ lowering the velocity of the wind and causing the 
snow carried by the wind to be deposited along the fence. 
Portable fences are frequently used, which are placed (by per- 
mission of the adjoining property owners) outside of the right- 
of-way. If a drift forms to the height of the portable fence the 
fence may be replaced on the top of the drift, where it may act 
as before, forming a still higher drift. When the prevailing 
wind runs along the track line, snow-fences built in short sec- 
tions on the sides will cause snow to deposit around them 
while it scours its way along the track line, actually clearing 
it. Such a method is in successful operation at some places on 
the White Mountain and Concord divisions of the Boston & 
Maine Railroad. Snow-fences, in connection with a moderate 
amount of shoveling and plowing, suffice to keep the tracks 
clear on railroads not troubled with avalanches. In such cases 
snow-sheds are the only alternative. 

291. Snow-sheds. These are structures which will actually 
keep the tracks clear from snow regardless of its depth outside. 
Fortunately they are only necessary in the comparatively rare 
situations where the snowfall is excessive and where the snow 
is liable to slide down steep mountain slopes in avalanches. 
These avalanches frequently bring dowm with them rocks, trees, 
and earth, which would otherwise choke up the road-bed and 
render it in a moment utterly impassable for weeks to come. 
The sheds are usually built of 12" X 12'' timber framed in about 
the same manner as trestle timbering; the ^'bents'' are some- 
times placed as close as 5 feet, and even this has proved insuffi- 
cient to withstand the force of avalanches. The sheds are there- 
fore so designed that the avalanche will be deflected over them 
instead of spending its force against them. Although these 
sheds are only used in especiall}^ exposed places, yet their length 
is frequentl}^ very great and they are liable to destruction by fire. 
To confine such a fire to a limited section, '' fire-breaks^' are 
made — i.e., the shed is discontinued for a length of perhaps 100 
feet. Then, to protect that section of track, a V-shaped de- 
flector will be placed on the uphill side which will deflect all 



§ 292. MISCELLANEOUS STRUCTURES AND BUILDINGS. 311 

descending material so that it passes over the sheds. SoHd crib 
work is largely used for these structures. Fortunately suitable 
timber for such construction is usually plentiful and cheap 
where these structures are necessary. Sufficient ventilation 
is obtained by longitudinal openings along one side immediately 
under the roof. "Summer'' tracks are usually built outside 
the sheds to avoid the discomfort of passing through these semi- 
tunnels in pleasant weather. The fundamental elements in 
the design of such structures is shown in Fig. 163, w^hich illus- 
trates some of the sheds used on the Canadian Pacific Railroad, 



12x15 




^^^0^^^^ 



Level-fall shed 
Fig. 163. — Snow-sheds — Canadian Pacific Railroad. 



292. Turn-tables. The essential feature of a turn-table is a 
carriage of sufficient size and strength to carry a locomotive, 
the carriage turning on a pivot of sufficient size to carry such a 
load. The carriage revolves in a circular pit w^hose top has 
the same general level as the surrounding tracks. The car- 
riages were formerly made largely of w^ood; very many of 
those still in use are of cast iron. Structural steel is now uni- 
versally employed for all modern work and since the construc- 
tion of the carriage and the pivot is a special problem in struc- 
tures, no further attention wdll here be paid to the subject 
except to that part which the railroad engineer must w^ork out 



312 RAILROAD CONSTRUCTION. § 292. 

— laying out the site and preparing the foundation. The 
minimum length of such a carriage (and therefore the diameter 
of the pit) is e\ddentl,y the length over all of the longest engine 
and tender in use on the road. Usually 60-foot turn-tables 
will suffice for an ordinary road, and for light-traffic roads 
employing small engines^ 50 feet or even less may be sufficient. 
Many of the heavier freight engines of recent make have a total 
length of about 65 feet; therefore 70-foot turn-tables are a 
better standard for heavy-traffic roads. A retaining-wall 
should be built around the pit. The stability of this wall imme- 
diately under the tracks should be especially considered. The 
most important feature is the stability of the foundation of the 
pivot, which must sustain a concentrated pressure, more or less 
eccentric, of perhaps 150 tons. When firm soil or rock may 
be easily reached, this need give no trouble, but in a soft, treach- 
erous soil a foundation of concrete or piling may be necessary. 
If the soil is very porous, it may be depended on to carry away 
all rain-water which may fall into the pit before the foundations 
are affected, but when .the soil is tenacious it may be necessary 
to drain the subsoil thoroughly and carry off immediately all 
surface drainage by means of subsoil pipes which have a suit- 
able outfall. 

The location of the turn-table in the yard is a part of the 
general subject of ^' Yards,'' and will be considered in the next 
chapter. 



CHAPTER XIII. 
YARDS AND TERMINALS, 

293. Value of proper design. A large part of the total cost of 
handling traffic, particularly freight, is that incurred at terminals 
and stations. In illustration of this, consider the relative total 
cost of handling a car-load of coal and a car-load (of equal 
weight) of mixed merchandise. The coal will be loaded in 
bulk on the cars at the mines, where land is comparatively 
cheap, and the cars grouped into a train without regard to order, 
since they are (usually) uniform in structure, loading, and con- 
tents. AMien the terminal or local station is reached they are 
run on tracks occupying property which is usually much cheaper 
than the site of the terminal tracks and freight-houses ; they are 
unloaded by gravity into pockets or machine conveyors and the 
empty cars are rapidly hauled by the train-Joad out of the w^ay. 
On the other hand, the merchandise is loaded by hand on the car 
from a freight-house occupying a central and valuable location, 
the car is hauled out into a yard occupying valuable ground, is 
drilled over the yard tracks for a considerable aggregate mileage 
before starting for its destination, w^here the same process is re- 
peated in inverse order. In either case the terminal expenses are 
evidently a large percentage of the total cost and, once loaded, 
it makes but little difference just how far the car is hauled to the 
other terminal. But the very evident increase in terminal charges 
for general merchandise over those for coal (large as they are) 
gives a better idea of the magnitude of terminal charges. 

Many yards are the result of gro^%i:h, adding a few" tracks at a 
time, without much evidence of any original plan. In such 
cases the yard is apt to be very inefficient, requiring a much 
larger aggregate of drilling to accomplish desired results, requir- 
ing much more time and hence blocking traffic and finally adding 
greatly to the cost of terminal service, although the fact of its 
being a needless addition to cost may be unsuspected or not fully 
appreciated. An unwillingness or inability to spend money for 

813 



314 RAILROAD CONSTRUCTION. § 294. 

the necessary changes, and the difficulty of making the changes 
while the yard is being used, only prolong the bad state of 
affairs and an inefficient makeshift is frequently adopted. As- 
sume that an improvement in the design of the yard will permit 
a saving of the use of one switching engine, or for example, that 
the work may be accomplished with three switching engines in- 
stead of four. Assuming a daily cost of $25, we have in 313 
working days an annual saving of $7825, which, capitalized at 
5%, gives $156,500, enough to reconstruct any ordinary yard.* 

294. Divisions of the subject. The subject naturally divides 
itself into three heads — -(a) Yards for receiving, classifying, and 
distributing freight cars, called more briefly freight yards; (b) 
yards and conveniences for the care of engines, such as ash tracks, 
turn-tables, coal-chutes, sand-houses, water-tanks, or water 
stand-pipes, etc., and (c) passenger terminals. 

FREIGHT YARDS. 

295. General principles. It should be recognized at the start 
that at many places an ideally perfect 3^ard is impossible, or at 
least impracticable, generally because ground of the required 
shape or area is practically unobtainable. But there are some 
general principles which may and should be followed in every yard 
and other ideals which should be approached as nearly as possi- 
ble. Nevertheless every yard is an independent problem. Be- 
fore taking up the design of freight yards, it is first necessary to 
consider the general object of such yards and the general princi- 
ples by which the object is accomplished. These may be briefly 
stated as follows: 

1. A yard is a device, a machine, by which incoming cars are 
sorted and classified — some sent to warehouses for unloading, 
some sent to connecting railroads, some made up for local dis- 
tribution along the road, some sent for repairs, and, in short a 
device by which all cars are sent through and out of the yard as 
quickly as possible. 

2. Except when a road's business is decreasing, or when its 
equipment is greater than its needs and its cars must be stored, 
efficiency of management is indicated by the rapidity with which 
the passage of cars through the yard is accomplished. 

3. When a yard is the terminal of a ^^ division," the freight 
— — — ■ ■ I 

* Estimate of Mr. H. G. Hetzler, C, B. & Q. Ry, 



§ 295. YARDS AND TERMINALS. 315 

trains will be pulled into a '' receiving track" and the engine and 
caboose detached. The caboose will be run on to a "caboose 
track/' which should be conveniently near, and the engine is run 
off to the engine yard. If the train is a " through" train and no 
change is to be made in its make-up, it will only need to wait for 
another engine and perhaps another caboose. If the cars are to 
be distributed, they will be drawn off by a switching engine to 
the "classification yard." 

4. The design of a j^ard is best studied by first picking out the 
ladder tracks and the through tracks which lead from one divi- 
sion of the yard to another. These are tracks which must always 
be kept open for the passage of trains, in contradistinction to 
the tracks on which cars may be left standing, even though it is 
onl}^ for a few moments, while drilling is being done. Such a set 
of tracks, which may be called the skeleton of the yard, is shown 
by heavy lines in Fig. 164. Each line indicates a pair of rails. 
The tracks of the storage yards are shown by the lighter lines. 

5. There is a distinct advantage in having all storage tracks 
double-ended — except "team tracks." Team tracks are those 
which have spaces for the accommodation of teams, so that load- 
ing or unloading may be done directly between the cars and teams. 
To avoid the necessity of teams passing over the tracks, these are 
best placed on the outskirts of the yard and consist of short stub- 
sidings arranged in pairs. But storage tracks should have an 
outlet at each end so as to reduce the amount of drilling neces 
sary to reach a car w^hich may be at the extreme end of a long 
string of cars. This is done usually by means of two "ladder" 
tracks, parallel to each other, which thus make the storage 
tracks betw^een them of equal length. 

6. The equality of length of these storage tracks is a point in- 
sisted on by many, but on the other hand, trains are not always of 
uniform length even on any one division. Loaded trains and 
trains of empties w^ill vary greatly in length, and the various 
styles and weights of freight engines employed necessitate other 
variations in the weights and lengths of trains hauled. With 
storage tracks of sLomewhat variable length a larger percentage 
of track length may be utilized, there will be less hauling over a 
useless length of track, and (assuming that the plot of ground 
available for yard purposes has equally favorable conditions for 
yard design) more business may be handled in a yard of given 
area. 



316 



RAILKOAD CONSTRUCTION. 



§295, 




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§ 296. YARDS AND TERMINALS. 317 

7 Yards are preferably built so that the tracks have a grade 
of 0.5% — sometimes a little more than this — in the direction of 
the traffic through the yard. This grade, which will overcome a 
tractive resistance of 10 pounds per ton, will permit cars to be 
started down the ladder tracks by a mere push from the switch- 
ing engine. They are then switched on to the desired storage 
track and run down that track by gravity until stopped at the 
desired place by a brakeman riding on the cars 

8. Although not absolutely necessary, there is an advantage 
in having all frog numbers and switch dimensions uniform. 
No. 7 frogs are most commonly used. Sharper-angled frogs 
make easier riding, less resistance and less chance of derailment, 
but on the other hand require longer leads and more space. No. 
G and even No. 5 frogs are sometimes used on account of economy 
of space, but they have the disadvantages of greater tractive re- 
sistance, greater wear and tear on track and rolling stock, and 
greater danger of derailment. 

296. Relation of yard to main tracks. Safety requires that 
there should be no connection between the yard tracks and the 
main tracks except at each end of the yard, where the switches 
should be ampl}^ protected by signals. Sometimes the main 
tracks run through the yard, making practically two yards— -one 
for the trafnc in either direction — but this either requires a double 
layout of tracks and houses (such as ash tracks, coal-chutes, sand- 
houses, etc.), or a very objectionable amount of crossing of the 
main- line tracks. The preferable method is to have the main hne 
tracks entirely on the outside of the j^ard. A method which is in 
one respect still better is to spread the main tracks so that they 
run on each side of the yard. In this case there is never any 
necessity to cross one main track to pass from the 3^ard to the 
other main track; a train may pass from the yard to either 
main track and still leave the other main track free and open. 
The ideal arrangement is that by which some of the tracks cross 
over or under all opposing tracks. By this means all connections 
between the 5^ard and the main tracks maybe by ^'trailing" 
switches; that is, trains will run on to the main track in the 
direction of motion on that main track. Of course all this 
applies only to double main track. 

An important element of yard design is to have a few tracks im- 
mediately adjoining the main tracks and separate from the yard 
proper on which outgoing trains may await their orders to take 



318 



RAILROAD CONSTRUCTION. 



§ 2G6. 




§ 297. YARDS AND TERMINALS. 319 

the main track. When the orders come, they may start at once 
without tiny delay, w ithout interfering with an}^ yard operations, 
and they are not occupying tracks which may form part of the 
system needed for switciiing. 

297. Minor freight yards. The term here refers to the sub- 
stations, only found in the largest cities, to which cars will be sent 
to save in the amount of necessary team hauling and also to re- 
lieve a congestion of such loading and unloading at the main 
freight terminal. The cars are brought to these yards sometimes 
on floats (as is done so extensively at various points around New 
York Harbor), or they are run dow^n on a long siding running 
perhaps through the city streets. But the essential feature of 
these yards is the maximum utilization of every square foot of 
yard space, which is alwa3^s very valuable and which is frequently 
of such an inconvenient shape that a great ingenuity is required 
to obtain good results. There is generally a temptation to use 
excessively sharp curves. When the radii are greater then 150 
feet no especial trouble is encountered. Curves with radius as 
short as 50 feet have been used in some yards. On such curves 
the long cars now generally used make a sharper angle wdth each 
other than that for which the couplers were designed and spe- 
cial coupler-bars become necessary. The two general methods 
of construction are (a) a series of parallel team tracks (as pre- 
^4ously described and as illustrated further in Fig. 165), and (b) 
the ''loop s^^stem," as is illustrated in Fig. 166. 

298. Transfer cranes. These are almost an essential feature 
for yards doing a large business. The transportation of built- 
up girders, castings for excessively heavy machinery, etc., which 
weigh five to thirty tons and even more, creates a necessity for 
machinery which will easily transfer the loads from the car to 
the truck smdvice versa. An ordinary ''gin-pole'' will serve the 
purpose for loads which do not much exceed five tons. A fixed 
framework, covering a span long enough for a car track and a 
team space, vnth sl trolley traveling along the upper chord, is the 
next design in the order of cost and convenience. Increasing 
the span so that it covers two car tracks and two team spaces 
will A^ery materially increase the capacity. Making the frame 
movable so that it travels on tracks which are parallel to the 
car tracks, giving the frame a longitudinal motion equal to two 
or three car lengths, and finally operating the raising and travel- 
ing mechanism by power, the facility for rapidly disposing of 



320 



RAILROAD CONSTRUCTION. 



§ 298. 




East 135th St. 

Fig. 166.— Minor Freight Yard on a Harbor Front. i 



§ 299. YARDS AND TERMINALS. 321 

heavy articles of freight is greatly increased. Of course only a 
very small proportion of freight requires such handling, and the 
business of a yard must be large or perhaps of a special character 
to justif}' and pay for the installation of such a mechanism. 
Figs. 165 and 166 each indicate a transfer crane, evidently of the 
fixed type. 

299. Track scales. The location of these should be on one of 
the receiving tracks near the entrance to the yard, but not on the 
main track. It is always best to have a ^'dead track" over the 
scales — i.e.. a track which has one rail on the solid side vv'all of 
the scale pit and the other supported at short intervals b}^ posts 
which come up through the scale platform and yet do not touch 
it. These rails and the regular scale rails switch into one track 
by means of point rails a few feet beyond each end of the scales. 
The switches should be normally set so that all trains will use 
the dead track, unless the scales are to be operated. It has been 
found possible in a gravity yard to weigh a train with very little 
loss of time by running each car slowly by gravity over the 
scales and weighing them as they pass over. 

ENGINE YARDS. 

300, General principles. Engine yards must contain all the 
tracks, buildings, structures, and facilities which are necessary 
for the maintenance, care, and storage of locomotives and for pro- 
viding them with all needed supplies. The supplies are fuel, 
water, sand, oil, waste, tallow, etc. Ash-pits are generally neces- 
sary for the prompt and economical disposition of ashes; engine- 
houses are necessary for the storage of engines and as a place 
where minor repairs can be quickly made. A turn-table is an- 
other all but essential req\iirement. The arrangement of all 
these facilities in an engine yard should properly depend on the 
form of the yard. In general the}^ should be grouped together 
and should be as near as possible to the place ^^'here through en- 
gines drop the trains just brought in and where they couple on 
to assembled outgoing trains, so that all unnecessary running light 
may be avoided. In Figs. 164 and 167 are shovrn two designs 
which should be studied with reference to the relative arrange- 
ment of the yard facilities. 



322 



RAILROAD CONSTRUCTION. 



§ 300, 




FiQ. 167. — Engine Yard and Shops, Urbana. III. 



i § 300. YARDS AND TERMINALS. 323 

I PASSENGER TERMINALS. 

(Passenger terminals are one of the logical subdivisions of 
this chapter, but their construction does not concern one engineer 
in a thousand. The local conditions attending their construction 
are so varied that each case is a special problem in itself — a prob- 
lem which demands in many respects the services of the archi- 
tect rather than the engineer. The student who wishes to pursue 
this subject is referred to an admirable chapter ir ^' Buildings and 
Structures of American Railroads/' by Walter G. Berg, Chief 
Engineer of the Lehigh Valley Railroad.) 



CHAPTER XIV. 

BLOCK SIGNALING. 
GENERAIi PRINCIPLES. 

301. Two fundamental systems. The growth of systems of 
block signaling has been enormous within the last few years — 
both in the amount of it and in the development of greater per- 
fection of detail. The development has been along two general 
lines : (a) the manual, in which every change of signal is the re- 
sult of some definite action on the part of some signalman, but in 
which every action is so controlled or limited or subject to 
the inspection of others that a mistake is nearly, if not quite, 
impossible; {h) the automatic^ in which the signals are oper- 
ated by mechanism, w^hich cannot set a wrong signal as long as the 
mechanism is maintained in proper order. The fundamental 
principles of the two systems will be briefly outlined, after 
which the chief details of the most common systems will be 
pointed out. 

302. Manual systems. Any railroad which has a telegraph 
line and an operator at all regular stations may (and generally 
does) operate its trains according to the fundamental princi- 
ples of the manual block system even though it makes no claim 
to a block-signal system. The basic idea of such a system is 
that after a train has passed a given telegraph- or signal-station, 
no other train will be permitted to follow it into that ^' block'' 
until word is telegraphed from the next station ahead that the 
first train has passed out of that block. With a double-track 
road the operation is very simple; trains may be run at short 
intervals wdth long blocks; with an average speed of 30 miles 
per hour and blocks 5 miles long, trains could be run on a ten 
minute interval (nearly). A road with any such traffic would, 
of course, have much shorter blocks, and, practically, they 
would need to be considerably shorter. 

With a single-track road the operation is much more complex, 
since the operator must keep himself informed of the move- 

324 



§ 303. BLOCK SIGNALING. 325 

ments of the trains in both directions. The ratio of length of 
block to train interval would l)e onh^ one half (and practically 
much less than half) what it could he with a double-track road 
When such a system is adhered to rigidly, it is called an absolute 
block system But when operating on this system, a delay of 
one train will necessarily delay every other train that follows 
closely after. A portion, if not all, of the delay to subsequent 
trains may be avoided, although at some loss of safety, by a 
system of permissive blocking. Ey this system an operator 
may give to a succeeding train a "clearance card" which per- 
mits it to pass into the next block, but at a reduced speed and 
with the train imder such control that it may be stopped on 
very short notice, especially near curves. One element of the 
danger of this sj^stem is* the discretionary power with which it 
invests the signalmen, a discretion which may be wrongfully 
exercised. A modification (which is a fruitful source of colli- 
sions on single-track roads) is to order two trains to enter a 
block approaching each other, and with instructions to pass 
each other at a passing siding at which there is no telegraph- 
station. When the instructions, are properly made out and 
literally obeyed, there is no trouble, but every thousandth or 
ten thousandth time there is a mistake in the orders, or a mis- 
understanding or disobedience, and a collision is the result. The 
telegraph line, a code of rules, a corps of operators, and sig- 
nals under the immediate control of the operators, are all that 
is absolutely needed for the simple manual system. 

303. Development of the manual system. One great diffi- 
culty with the simple system just described is that each operator 
is practically independent of others except as he may receive 
general or specific orders from a train-dispatcher at the division 
headquarters. Such difficulties are somewhat overcome by a 
very rigid system of rules requiring the signalmen at each station 
to. keep the adjacent signalmen or the train-dispatcher in- 
formed of the movements of all trains past their own stations. 
When these rules (which are too extensive for quotation here) 
are strictly observed, there is but little danger of accident, and 
a neglect by any one to observe any rule will generally be appar- 
ent to at least one other man. Nevertheless the safety of trains 
depends on each signalman doing his duty, and a little careless- 
ness or forgetfulness on the part of any one man may cause an 
accident. The signaling between stations may be done by 



326 RAILROAD CONSTRUCTION. § 303. 

ordinary telegraphic messages or by telephone, but is frequently 
done by electric bells, according to a code of signals, since these 
may be readily learned by men who would have more difficulty 
in learning the Morse code. 

In order to have the signalmen mutually control each other, 
the ''controlled manuaP' system has been devised. The first 
successful system of this kind which was brought into exten- 
sive use is the ^'Sykes'^ system, of which a brief description 
is as follows: Each signal is worked by a lever; the lever is 
locked by a latch, operated by an electro-magnet, which, with 
other necessar}^ apparatus, is inclosed in a box. When a signal 
is set at danger, the latch falls and locks the lever, which cannot 
be again set free until the electro-magnet raises the latch. The 
magnet is energized only by a current, the circuit of which is 
closed by a ''plunger" at the next station ahead; just above 
the plunger is an "indicator," also operated by the current, 
which displays the words clear or blocked. (There are varia- 
tions on this detail.) When a train arrives at a block station 
(A), the signalman should have previously signaled to the station 
ahead (B) for permission to free the signal. The man ahead (B) 
pushes in the "plunger" on his instrument (assuming that the 
previous train has already passed him), which electricall}^ opens 
the lock on the lever at the previous station {A). The signal 
at A can then be set at "safety." As soon as the train has 
passed A the signal at A must be set at " danger." A further 
development is a device by which the mere passage of the train 
over the track for a few feet beyond the signal will automati- 
cally throw the signal to "danger." After the signal once goes 
to danger, it is automatically locked and cannot be released 
except by the man in advance (B), who will not do so until the 
train has passed him. The "indicator" on B^s instrument 
shows "blocked" when A's signal goes to danger after the train 
has passed A, and B's plunger is then locked, so that he can- 
not release A's signal while a train is in the block. As soon as 
the train has pa ssed A , B should prepare to get his signals ready 
by signaling ahead to C, so that if the block between B and C 
is not obstructed, B may have his signals at "safety" so that 
the train may pass B without pausing. The student should 
note the great advance in safety made by the Sykes system; 
a signal cannot be set free except by the combined action of 
two pnien^ one the man who actually operates the signal and 



§ 304. BLOCK SIGNALING. 327 

the other the man at the station ahead, who frees the signal 
electrically and who by his action certifies that the block im- 
mediately ahead of the train is clear. 

A still further development makes the system still more '' auto- 
matic" (as described later), and causes the signal to fall to dan- 
ger or to be kept locked at danger, if even a single pair of wheels 
comes on the rails of a block, or if a switch leading from a main 
track is opened. 

304. Permissive blocking. ^'Absolute" blocking renders ac- 
cidents due to collisions almost impossible unless an engineer 
runs by an adverse signal. The signal mechanism is usually 
so designed that, if it gets out of order, it will inevitably fall to 
^'danger," i.e., as described later, the signal-board is counter- 
balanced by a weight which is much hea^der. If the wire breaks, 
the counterw^eight will fall and the board will assume the hori- 
zontal position, which always indicates ''danger.'^ But it some- 
times happens that when a train arrives at a signal-station, the 
signalman is unable to set the signal at safety. This may be 
because the previous train has broken down somewhere in the 
next block, or because a switch has been left open, or a rail has 
become broken, or there is a defect of some kind in the electrical 
connections. In such cases, in order to avoid an indefinite 
blocking of the whole traffic of the road, the signalman may 
give the engineer a ^'caution-card" or a '^ clearance card," 
which authorizes him to proceed slowly and with liis train imder 
complete control into the block and through it if possible. If 
he arrives at the next station without meeting any obstruction 
it merely indicates a defective condition of the mechanism, 
which will, of course, be promptly remedied. Usually the next 
section will be found clear, and the train may proceed as usual. 
On roads where the '^ controlled manual" system has received 
its highest development, the rules for permissive blocking are 
so rigid that there is but little danger in the practice, unless 
there is an absolute disobedience of orders. 

305. Automatic systems. By the verj^ nature of the case, 
such systems can only be used to indicate to the engineers of 
trains something with reference to the passage of previous 
trains. The complicated shifting of switches and signals which 
is required in the operation of yards and terminals can only be 
accomplished by '^ manual" methods, and the only automatic 
features of these methods consist in the mechanical checks 



328 RAILROAD CONSTRUCTION. § 306. 



in 



01 
-.)Yl 



(electric and otherwise), which will prevent wrong combina- 
tions of signals. But for long stretches of the road, where it f[ 
is only required to separate trains by at least one block length, 
an automatic system is generally considered to be more relia- ^^ 
ble. As expressed forcibly by a railroad manager, "an auto- 
matic system does not go to sleep, get drunk, become insane, 
or tell lies when there is any trouble." The same cannot always 
be said of the employes of the manual system. 

The basic idea of all such systems is that when a train pi 
a signal-station (A), the signal automatically assumes the " r* 
ger" position. This may be accomplished electrically, p: ^ 
matically, or even b}^ a direct mechanism. When the t 
reaches the end of the block at B and passes into the next » :E 
the signal at B will be set at danger and the signal at A wil ': ^ 
set at safety. The lengths of the blocks are usually so gi » » 
that the only practicable method of controlling from i x- 
mechanism at A is by electricity, although the actual mol -i 
power at A may be pneumatic or mechanical. At one ti il 
the current from A to B was carried on ordinar}^ wires. T > ' 
method has the very positive advantage of reliability, defir ,i 
resistance to the current, and small probability of short-circi •> ' 
ing or other derangement. But now all such systems use t -.y: 
rails for a track circuit and this makes it possible to detect i ij 
presence of a single pair of wheels on the track anywhere in t [o 
block, or an open switch, or a broken rail. Any such circm /, 
stances, as well as a defect in the mechanism, will break 
short-circuit the current and will cause the signal to be set i 
danger. To prevent an indefinite blocking of traffic owing '< 
a signal persistently indicating danger, most roads employii 
such a system have a rule substantially as follows: When a trai . 
finds a signal at danger, after waiting one minute (or mor< 
depending on the rules), it may proceed slowly, expecting t 
find an obstruction of some sort; if it reaches the next bloc 
without finding any obstruction and finds the next signal clear, 
it may proceed as usual, but must promptly report the case to 
the superintendent. Further details regarding these methods 
will be given later. See § 310. 

306. "Distant" signals. The close running of trains that 
is required on heavy-traffic roads, especially where several 
branches combine to enter a common terminal, necessitates the 
use of ver}^ short l^locks. A heavy train running at high speed 



f: 



307. BLOCK SIGNALING. 329 

stop in less than 2000 feet, ^vhile 

e curves of a road (or other obstructions) frequently make 

i difficult to locate a signal so that it can be seen more than a 

^bw hundred feet away. It would therefore be impracticable 

|lo maintain the speed now used with heavy trains if the engi- 

leer had no foreknowledge of the condition in which he will 

|nd a signal until he axrives within a short distance of it. To 

overcome this difficulty the "distant signal was devised. This 

.^ced about 1800 or 2000 feet from the ^4iome'^ signal, and 

,[ .., .?ierloeked with it so that it gives the same signal. The dis- 

signal is frequently placed on the same pole as the home 

.1 of the previous block. When the engineer finds the 

ic .nt signal "clear/' it indicates that the succeeding home 

[il is also clear, and that he may proceed at full speed and 

^expect to be stopped at the next signal; for the distant 

^ul cannot be cleared until the succeeding home signal is 

red, which cannot be done until the block succeeding that 

^, ear. A clear distant signal therefore indicates a clear track 

two succeeding blocks. When the engineer finds the distant 

j:,al blocked, he need not stop (providing the home signal is 

t). It simply indicates that he must be prepared to stop 

:he next home signal and must reduce speed if necessary. 

nay happen that by the time he reaches the succeeding home 

lal it has already been cleared, and he may proceed without 

(ipping. This device facilitates the rapid running of trains, 
h no loss of safety, and yet with but a moderate addition to 
signaling plant. 
,07. "Advance " signals. It sometimes becomes necessary 
locate a signal a fevv hundred feet short of a regular passen- 
-station. A train might be halted at such a signal because 
was not cleared from the signal-station ahead — perhaps a 
fie or two ahead. For convenience, an " advance '' signal 
:ij be erected immediately beyond the passenger-staticn. 
ine train will then be permitted to enter the block as far as 
the advance signal and may deliver its passengers at the station. 
The advance signal is interlocked with the home signal back 
of it, and cannot be cleared until the home signal is cleared and 
the entire block ahead is clear. In one sense it adds another 
block, but the signal is entirely controlled from the signal station 
back of it. 



330 EAILROAD CONSTRUCTION. § 308. 



MECHANICAL DETAILS. 

308. Signals. The primitive signal is a mere cloth flag. A 
better signal is obtained when the flag is suspended in a suit- 
able place from a fixed horizontal support, the flag weighted 
at the bottom, and so arranged that it may be drawn up and 
out of sight by a cord which is run back to the operator's office. 
The next step is the substitution of painted wood or sheet metal 
for the cloth flag, and from this it is but a step to the standard ^ 
semaphore on a pole, as is illustrated in Fig. 168. The simple 
flag, operated for convenience with a cord, is the signal em- 
ployed on thousands of miles of road, where they perhaps make ' 
no claim to a block-signal system, and yet where the trains 
are run according to the fundamental rules of the simple manual 
block method. 

Semaphore boards. These are about 5 feet long, 8 inches 
wide at one end, and tapered to about 6 inches Avide at the hinge 
end. The boards are fastened to a casting which has a ring to 
hold a red glass which may be swung over the face of a lantern, 
so as to indicate a red signal. ^'Distant" signal-boards usually 
have their ends notched or pointed; the ^'home'' signal-boards 
are square ended. The boards are always to the right of the 
hinge when a train is approaching them. The ''home" signigls 
are generally painted red and the '^ distant" signals green, 
although these colors are not invariable. The backs of t$\8 
boards are painted white. Therefore any signal-board whi 1 
appears on the left side of its hinge will also appear white j and , 
is a signal for traffic in. the opposite direction, and is therefore] 
of no concern to an engineman. ' 

Poles and bridges. When the signals are set on poles, they 
are generall}^ placed on the right-hand side of the track. When 
there are several tracks, four or more, a bridge is frequently 
built and then each signal is over its own track. When switches 
run off from a main track, there may be several signal-boards 
over one track. The upper one is the signal for the main track 
and the lower ones for the several switches. In Fig. 169 is \ 
shown a '^ bridge" with its various signal-boards controlHng the i 
several tracks and the switches running off from them. 

"Banjo" signals. This name is given to a form of signal, 
illustrated in Fig. 170, in which the indication is taken from the 



{To face page 330.) 




Fig. 168. — Semaphore«. 



ill 



{To face 'page 330.) 





X 



^1 



\,, 





■■\ 



X 



Fig. 170. — "' Banjo '' Signals. 



§ 309. BLOCK SIGNALING. • 331 

color of a round disk inclosed with glass. This is the distinctive 
signal of the Hall Signal Company, and is also used by the 
Union Switch and Signal Company. The great argument in 
.j their favor is that they may be worked by an electric current 
|i of low voltage, which is therefore easily controlled; that the 
•I mechanism is entirely inside of a case, is therefore very light, 
|l and is not exposed to the weather. Ihe argument urged 
I against them is that it is a signal of color rather than form 
jj or position, and that in foggy ^Acather the signal cannot be 
seen so easily; also that unsuspected color-blindness on the 
I part of the engineman may lead to an accident. Notwith- 
standing these objections, this form of signal is used on thousands 
of miles of line in this country. 
I 309. Wires and pipes. Signals are usually operated by levers 
in a signal-cabin, the levers being very similar to the reversing- 
lever of a locomotive. The distance from the levers to the sig- 
nals is, of course, very variable, but it is sometimes 2000 feet. 
The connecting-link for the most distant signals is usually 
No. 9 wire; for nearer signals and for all switches operated 
from the cabin it may be 1-inch pipe. When not too long, one 
pipe will serve for both motions, forward and back. When 
wires are used, it is sometimes so designed (in the cheaper sa^s- 
tems) that one wire serves for one motion, gravity being de- 
pended on for the other, but now all good systems require two 
wires for each signal. 

Compensators. Variations of temperature of a material with 
as high a coefficient as iron wall cause very appreciable differ- 
ence of length in a distance of several hundred feet, and a 
dangerous lack of adjustment is the result. To illustrate: A 
fall of 60^ F. will change the length of 1000 feet of wire by 

1000 X 60 X. 0000065 = 0.39 foot =4.68 inches. 

A much less change than this will necessitate a readjustment 
of length, unless automatic compensators are used. A com- 
pensator for pipes is very readil}^ made on the principle illus- 
trated in Fig. 171. The problem is to preserve the distance 
between a and d constant regardless of the temperature. Place 
the compensator half-way between a and d, or so that ab=cd. 
A fall of temperature contracts ab to ab\ Gloving b to y will 
cause c to move to c', in which bb^ =cc\ But cd has also short- 
eiied to c'd; therefore d remains fixed in position. To avoid 



332 



RAILROAD CONSTRUCTION. 



§ 310.1 



too great angular motion, one such compensator should be used 
for each 500 feet. If a line 1000 feet long is to be provided 
for, two compensators would be used, 250 feet from each end. 
Note that in operating through a compensator the direction 
of motion changes; i.e., if a moves to the right, d moves to the 
left, or if there is compression in ah there is tension in cd, and 





Fig. 171. — Standard Pipe Compensator. 

vice versa. Therefore this form of compensator can onh^ be 
used with pipes which will withstand compression. It has ' 
seemed impracticable to design an equally satisfactory^ com- i 
pensator for wires, although there are several designs on the 
market. 1 

Guides around curves and angles. When wires are required i 
to pass around curves of large angle, pulleys are used, and a s 
length of chain is substituted for the wire. For pipes, when ] 
the curve is easy the pipes are slightly bent and are guided I 
through pulleys. When the angle is sharper, '' angles'' are 
used. The operation of these details is self-evident from an 
inspection of Fig. 172. 

310. Track circuit for automatic signaling. The several 
systems of automatic signaling differ in the minor details, but 1 



§310. 



BLOCK SIGNALING. 



333 



nearly all of them agree in the following particulars. A current 
of low potential is run from a l^attery at one end of a section 
through one line of rails to the other end of tlie section, tlien 
through a relay, and then back to the battery through the other 




Fig. 172. — Deflecting-rods. 



line of rails. To avoid the excessive resistance which would 
occur at rail joints w^hich may become badly rusted, a wire 
suitably attached to the rails is run around each joint. In 
order to insulate the rails of one section from the rails at either 
end and yet maintain the rails structurally continuous, the 
ends of the rails at these dividing points are separated by an 
insulator and the joint pieces are either made of wood or have 
some insulating material placed between the rails and the ordi- 
nary metal joint. The bolts must also be insulated. When 
the relay is energized by a current, it closes a local circuit at 
the signal-station, which will set the signal there at ''safety.'^ 
The resistance of the relay is such that it requires nearly the 
whole current to work it and to keep the local circuit closed. 
Therefore, -vA'hen there is any considerable loss of current from 
one rail to the other, the relay will not be sufhcientlv energized, 
the local circuit will be broken, and the signal will automatically 
k fall to danger. This diversion of current from one rail to the 
other before the current reaches the relay may be caused in 
several ways: the presence of a pair of wheels on the rails any- 
where in the section will do it; also the breakage of a rail; also 
the opening of a switch anywhere in the section ; also the pres- 
ence of a pair of wheels on a siding between the ^'fouling point'' 
^ and the switch. (The ''fouhng point" of a siding is that point 
I where the rails first commence to approach the main track.) 
I In Fig. 173 is shown all of the above details, as well as some 
' others. At A, B, and the ''fouling point" are shown the in- 



I 



334 



RAILROAD CONSTRUCTION. 



§ 310, 



w E 



sulated joints. The batteries and signals are arranged for 
_^ train motion to the right. When a 

train has passed the points near A, 
where the wires leave the rails for 
the relay, the current from the "track 
battery" at B will pass through the 
wheels and axles, and although no 
electrical connection is broken, so 
much current will be shunted through 
the wheels and axles that the weak 
current still passing through the relay 
is not strong enough to energize it 
against its spring and the "signal- 
magnet" circuit is broken, and the 
signal A goes to "danger." At the 
turnout the rails between the foul- 
ing point and the switch are so con- 
II °'li >^ nected (and insulated) that a pair of 

\ wheels on these rails will produce the 
same effect as a pair on the main 
track. This is to guard against the 
effect of a car standing too near the 
switch, even though it is not on the 
main track. When the train passes 
B, if there is no other interruption 
of the current, the track battery at 
B again lenergizes the relay at A, 
the signal-magnet circuit at A is 
closed, and the signal is drawn to 
"safety." 

(The present edition has omitted 
several subdi^asions of this general 
subject, notably the "staff system," 
used chiefly in England, and all dis- 
cussions of "interlocking" which is 
an essential feature of the opera- 
tion of large terminal yards. A future 
edition may supply these deficiencies, 
although an exhaustive treatment of 
the subject of Signaling would require 
Fig. 173. a separate volume.) 



Il<:|| 




CHAPTER XV. 



ROLLING-STOCK. 



(It is perhaps needless to say that the following chapter is 
in no sense a course in the design of locomotives and cars. Its 
chief idea is to give the student the elements of the construc- 
tion of those vehicles which are to use the track which he may 
design — to point out the mutual actions and reactions of vehicle 
against track and to show the effect on track wear of varia- 
tions in the design of rolling-stock. The most of the matter 
given has a direct practical bearing on track-work, and it is con- 
sidered that all of it is so closely related to his work that the 
civil engineer may study it with profit.) 

WHEELS AND RAILS. 

311. Effect of rigidly attaching wheels to their axles. The 
wheels of railroad rolling-stock are invariably secured rigidly 
to the axles, which therefore revolve with the wheels. The 
chief reason for this is to avoid excessive wear 
between the axles and the wheels. 

Any axle must always be somewhat loose in 
its journals. A sidewise force P (see Fig. 174) 
acting against the circumference of the wheel 

I will produce a much greater pressure on the 

I axle at S and S^, and if the wheel moves on 

;the axle, the wear at S and S^ will be exces- 
sive. But when the axle is fitted to the wheel 
with a ''forced fit" and does not revolve, 
the mere pressure produced at iS> is harmless. 
When two wheels are fitted tight to an axle, 
as in Fig. 175, and the axle revolves in the jour- Fig. 174. 

nals an, a sidewise pressure of the rail against the wheel flange 
will only produce a slight and harmless increase of the journal 
pressure Q, although at Q there is sliding contact. Twist- 

i 335 




336 



RAILROAD CONSTRUCTION. 



§ 312. 



ing action in the journals is thus practically avoided, since a 
small pressure at the journal-boxes at each end of the axle 
suffices to keep the axle truly in line. - 



Fig. 175. 



.-n' 






-n. 



■2~^ 




Fig. 176. 



On the other hand, when the wheels are rigidly attached to 
their axles, both wheels must turn together, and when rounding 
curves, the inner rail being shorter than the outer rail, one 
wheel must slip by an amount equal to that difference of length. 
I'he amount of this slip is readily computable : 



Longitudinal slip 



-T.^O 



27ra 



271(7 



360 



o(r2 ^i)-35oc 



Ca' 



(136) 



in which C is a constant for any one gauge, and g= the track 
gauge = (r2 — rj . For standard gauge (4.708) the shp is .08218 
foot per degree of central angle. This shows that the longitu- 
dinal slipping around any curve of any given central angle will 
be independent of the degree of the curve. The constant (.08218) 
here given is really somewhat too small, since the true gauge 
that should be considered is the distance between the lines of 
tread on the rails. This distance is a somewhat indeterminate 
and variable quantity, and probably averages 4.90 feet, Avhich 
would increase the constant to .086. The slipping may occur 
by the inner wheel slipping ahead or the outer wheel slipping 
back, or by both wheels slipping. The total slipping will be 
constant in any case. The slipping not only consumes power, 
but wears both the wheels and the rail. But e^^en these dis- 
advantages are not sufficient to offset the advantages resulting 
from rigid wheels and axles. 

312. Effect of parallel axles. Trucks are made Avith two or 
three parallel axles (except as noted later), in order that the 
axles shall nuitually guide each other and be kept approximately 



m 



§312. 



ROLLING-STOCK. 



337 



(| perpendicular to the rails. If the curvature is very sharp and 

' the wheel-base comparatively long (as is notably the case on 

street railwavs at street corners), the front and rear wheels 





Fig. 177. 



Fig. 178. 



^r^ 







Fig. 179. 



wUl stand at the same angle (a) with the track, as shown in 
Fig. 177. But it has been noticed that for ordinary degrees of 
curvature, the rear, wheels stand radial to the curve (see Fig. 
178), and for steam railroad w^ork this is the normal case. When 
the two parallel axles are on a curve (as shown), the ^^'heels tend 
to run in a straight line. In order that they shall run on a curve 
they must slip laterally. The principle 
is illustrated in an exaggerated form in 
Fig. 179. The wheel tends to roll from a ^- 
toward h. Therefore in passing along the 
track from a to c it must actually slip late- ''" 
rally an amount he which equals ac sin a. 
I.et ^=length of the wheel-base (Figs. 177 and 178); r = radius 
of curve; then for the first case (Fig. 177), sina = /-^2r; for 
the second and usual case (Fig. 178), sin a = ^-^r; for ^=5 feet 
and r = radius of a 1° cmwe, a=-0°03' for the second case, a 
varies (practically) as the degree of curve. The lateral slipping 
yer unit of distance traveled therefore equals sin a. As an 
illustration, given a 5-foot wheel-base on a 5° curve, a = 0° 15', 
sin a = .00436, and for each 100 feet traveled along the curve 
the lateral slip of the front wheels would be 0,436 foot. There 
would be no lateral slipping of the rear wheels, assuming that 
the rear axle maintained itself radial. 

From the above it might h^. inferred that the flanges of the 
forward wheels will have much greater wear than those of the 
rear wheels. Since cars are drawn in both directions about 
equally, no difference in flange wear due to this cause will occur, 
but locomotives (except switching-engines) run forward almost 



338 



KAILROAD CONSTRUCTION. 



§313. 



m 



exclusively, and the excess wear of the front wheels of the pilot - 
and tender-trucks is plainly observable. 

For a given curve the angle a (and the accompanying resist- 
ance) is evidently greater the greater the distance between 
the axles. On the other hand, if the two axles are very close 
together, there will be a tendency for the truck to twist and 
the wheels to become jammed, especially if there is consider- 
able play in the gauge. The flange friction would be greater 
and would perhaps exceed the saving in lateral slipping. A 
general rule is that the axles should never be closer together 
than the gauge. 

Although the slipping per unit of length along the curve varies 
directly as the degree of curvature, the length of curve necessary 
to pass between two tangents is inversely as the degree of curve, 
and the total slipping between the two tangents is independent 
of the degree of curve. Therefore when a train passes between 

two tangents, the total slipping 
of the wheels on the rails, lon- 
gitudinal and lateral, is a quantity 
which depends only on the central 
angle and is independent of the 
radius or degree of curve. 

313. Effect of coning wheels. 
The wheels are always set on the 
axle so that there is some ^^play'^ 
or chance for lateral motion be- 
ij j j I: tween the wheel-flanges and the 

^pU I 1 [i-"^ rail. The treads of the wheel are 

I T I also '^ coned.'' This coning and play 

Pj^ ^gQ of gauge are shown in an exagger- 

ated form in Fig. 180. When the 
wheels are on a tangent, although there will be occasional oscil- 
lations from side to side, the normal position will be the sym- 
metrical position in Avhich the circles of tread hh are equal. 
When centrifugal force throws the wheel-flange against the rail, 
the circle of tread a is larger than 6, and much larger than c; 
therefore the wheels will tend to roll in a circle whose radius 
equals the slant height of a cone whose elements would pass 
through the unequal circles a and c. If this radius equaled the 
radius of the track, and if the axle were free to assume a radial 
position, the wheels would roll freely on the rails without any 



„^rxi^.„ 



■H^ 






i § 314. ROLLING-STOCK. 339 

j slipping or flange pressure. Under such ideal conditions, 
coning would be a valuable device, but it is impracticable to 
have all axles radial, and the radius of curvature of the track 
is an extremely variable quantity. It has been demonstrated 
that with parallel axles the influence of coning diminishes as 

i the distance between the axle increases, and that the effect is 
practically inappreciable when the axles are spaced as they are 
on locomotives and car-trucks. The coning actuall}^ used is 
very slight (see Chapter XV, § 332) and has a different object. 
It is so slight that even if the axles were radial it would only 

I prevent the slipping on a very light curve — say a 1° curve. 

314. Effect of flanging locomotive driving-wheels. If all the 
wheels of all locomotives were flanged it would be practically 
impossible to run some of the longer types around sharp curves. 
The track-gauge is always widened on curves, and especially 
on sharp curves, but the widening would need to be excessive 
to permit a consolidation locomotive to pass around an 8° or 
10° curve if all the drivers were flanged. The action of the 

' wheels on a curve is illustrated in Figs. 181, 182, and 184. AU 
small truck-Tvheels are flanged. The rear drivers are always 
flanged and four-driver engines usually have all the drivers 
flanged. Consolidation engines have only the front and rear 
drivers flanged. Mogul and ten-wheel engines have one pair 
of drivers blank. On Mogul engines it is always the middle 
pair. On ten- wheel engines, ^^hen uced on a road having sharp 
curves, it is preferable to flange the front and rear dri^dng- 
wheels and use a ^' swing bolster" (see § 315); Avhen the curva- 
ture is eas}^, the middle and rear drivers may be flanged and 
the truck made with a rigid center. The blank drivers have 
the same total width as the other drivers and of course a much 
wider tread, which enables these drivers to remain on the rail, 
even though the curvature is so sharp that the tread overhangs 
the rail considerably. 

315. Action of a locomotive pilot-truck. The purpose of 
the pilot-truck is to guide the front end of a locomotive around 
a curve and to relieve the otherwise excessive flange pressure 
that would be exerted against the driver-flanges. There are 
two classes of pilot-trucks — (a) those having fixed centers and 
(h) those having shifting centers. This second class is again 
subdivided into two classes, which are radically different in 
their action — (b^) four-wheeled trucks having two parallel axles 



340 



RAILROAD CONSTRUCTION, 



§315. 



and (b2) two-wheeled trucks which are guided by a ^'radius- 
bar." The action of the four-wheeled fixed-centered truck (a) 
is sho\\Ti in Fig. 181. Since the center of the truck is forced 




Fig. 181. — Fixed Center Pilot-truck. 
to be in the center of the track, the front drivers are drawn 
away from the outer rail. The rear outer driver tends to roll 
away from the outer rail rather than toward it, and so the effect 




Fig. 182. — Four-wheeled Truck — Shifting Center. 

of the truck is to relieve the driver-flanges of an}^ excessive 
pressure due to curvature. The only exception to this is the 
case where the curvature is sharp. Then the front inner driver 
may be pressed against the inner rail, as indicated in Fig. 181. 

This limits the use of this type of 
wheel-base on the sharper curves. 
The next type — (b^) four-wheeled 
trucks with shifting centers — is 
much more flexible on sharp 
curvature; it likewise draws the 
front drivers away from the outer 
rail. The relative position of the 
wheels is shown in Fig. 182, in 
which c' represents the position 
of center-pin and c the displaced 
truck center. The structure and 
action of the truck is shown in 
Fig. 183. The '^ center-pin" (1) is 
supported on the ''truck-bolster" (2), w^hich is hung by the 
''links" (4) from the "cross-ties" (3). The links are therefore 




Fig. 183. 



-Action of Shifting 
Center. 



f 315. 



ROLLING-STOCK. 



341 



in tension and when the wheels are forced to one side by the 
rails the links are inclined and the front of the engine is 
drawn inward by a force equal to the weight on the bolster 
times the tangent of the angle of incHnation of the links. This 
assumes that all links are vertical when the truck is in the 
center. Frequently the opposite links are normally inclined to 
each other, which somewhat complicates the above simple relation 
of the forces, although the general principle remains identical. 

The two-wheeled pilot-truck with shifting center is illus- 
trated in Fig. 184. The figure shows the facility with which 





Fig. 184. — Two-wheeled Truck — Shifting Center. 
an engine with long wheel-base may be made to pass around 
a comparatively sharp curve by omitting the flanges from the 
middle drivers and using this form of pilot-truck. As in the 
previous case, the eccentricity of 
the center of the truck relative 
to the center-pin induces a cen- 
tripetal force which draws the 
front of the engine inward. But 
the swing- truck is not the only 
source of such a force. If the 
"radius-bar pin'' were placed at O' (see Fig. 185), the truck- 
axle would be radial. But the radius-bar is always made some- 
what shorter than this, and the pin is placed at 0, a considerable 
distance ahead of 0', thus creating a tendency for the truck 
to run toward the inner rail and draw the front of the loco- 
motive in that direction. This tendency will be objectionably 
great if the radius-bar is made too short, as has }>een practically 
demonstrated in cases when the radius-bar has been subse- 
quently lengthened with a resulting improvement in the running 
of the engine. 



Fig. 185. — Action of Two- 
wheeled Truck. 



342 RAILROAD CONSTRUCTION. § 316. 

LOCOMOTIVES. 
GENERAL STRUCTURE. 

316. Frame. The frame or skeleton of a locomotive con- 
sists chiefly of a collection of forged wrought-iron bars, as 
shown in Figs. 186 and 187. These bars are connected at the 



4 




Fig. 186. — Engine-frame. 

front end by the "bumper" (c), which is usually made of wood. 
A httle further back they are rigidly connected at bh by the 
cylinders and boiler-saddle. The boilers rest on the frames 
at aaaa by means of ''pads/' which are bolted to the fire-box, 
but which permit a free expansion of the boiler along the frame. 
This expansion is sometimes as much as yV'. On a ''con- 
solidation" engine (frame shown in Fig. 187) it is frequently 




EEflflaDiP^ 



i 

Fig. 187. — Engine-frame — Consolidation Type. 

necessary to use vertical swing-levers about 12'' long instead 
of "pads." The swinging of the levers permat all necessary 
expansion. At the back the frames are rigidly connected by 
the iron "foot-plate." The driving-axles pass through the 
"jaws" dddd, which hold the axle-boxes. The frame-bars 
have a width (in plan) of 3" to 4''. The depth (at a) is about 
the same. Fig. 186 shows a frame for an "American" type 
of locomotive; Fig. 187 shows a frame for a " Consolidation" 
type (see § 323). 

317. Boiler. A boiler is a mechanism for transferring the 
la.tent heat of fuel to water, so that the water is transformed 
from cold water into high-pressure steam, which by its expan- 
sion will perform work. The efficiency of the boiler depends 
largely on its ability to do its work rapidly and to reduce to 
a minimum the w^aste of heat through radiation. The boiler 
contains a fire-box (see Fig. 188), in which the fuel is burned. 
The gases of consumption pass from the fire-box through the 
numerous boiler-tubes into the ''smoke-box" ;S and out through 
the smoke-stack. The fire-box consists of an inner and outer 



§ 317. 



ROLLING-STOCK. 



343 



shell separated by a layer of water about 3" thick. The ex- 
posure of water-snrfaee to the influence of the fire is thus very 
complete. The efficiency of this transferal of heat is somewhat 
indicated by the fact that, although the temperature of the 
gases in the fire-box is probably from 3000° to 4000° F., the 
temperature in the smoke-box is generally reduced to 500° to 




Fig. 188. — Locomotive -boiler. 
600° F. If the steam pressure is 180 lbs., the temperature of 
the water is about 380° F., and, considering that heat will not 
pass from the gas to the water unless the gas is hotter than the 
water, the water evidently absorbs a large part of the theo- 
retical maximum. Nevertheless gases at a temperature of 
600° F. pass out of the smoke-stack and such heat is utterly 
wasted. 

The tubes vary from If to 2" , inside diameter, with a thick- 
ness of about O'MO to 0'M2= The aggregate cross-sectional 
area of the tubes should be about one eighth of the grate area. 
The number will vary from 140 to 250. They are made as long 
as possible, but the length is ^drtually determined by the tj^pe 
and length of engine. 

318. Fire-box. The fire-box is surroimded by water on the 
four sides and the top, but since the water is subjected to the 
boiler pressure, the plates, which are about -l~' thick, must be 
stayed to prevent the fire-box from collapsing. This is easily 
accomplished over the larger part of the fire-box surface by 
having the outside boiler-plates parallel to the fire-box plates 
and separated from them by a space of about 3". The plates 
are then mutually held by ''stay-bolts." See Fig. 189. These 
are about \" in diameter and spaced 4" to \y , The f^' hole, 



344 



RAILROAD CONSTRUCTION. 



§318. 



drilled li'' deep, indicated in the figure, will allow the escape 
of steam if the bolt breaks just behind the plate, and thus calls 
attention to the break. The stay-bolts are turned down to a 
diameter equal to that at the root of the screw-threads. This 
method of supporting the fire-box sheets is used for the two 
sides, the entire rear, and for the front of the fire-box up to the 
boiler-barrel. The '' furnace tube-sheet" — the upper part of 
the front of the fire-box — is stayed by the tubes. But the top 
of the fire-box is troublesome. It must always be covered 
with w^ater so that it will not be ^'burned" by the intense heat. 
It must therefore be nearly, if not quite, flat. There are three 
general methods of accomplishing this. 




Fig. 189. 



Fig. 190. 



(a) Radial stays. This construction is indicated in Fig. 190. 
Incidentally there is also shown the diagonal braces for resist- 
ing the pressure on the back end of the boiler above the fire- 
box. It may be seen that the stays are not perpendicular to 
either the crown-sheet or the boiler-plate. This is objection- 
able and is obviated by the other methods. 

(b) Crown-bars. These bars are in pairs, rest on the side 
furnace-plates, and are further supported by stays. See Fig. 
191. 

(c) Belpaire fire-box. The boiler above the fire-box is rect- 
angular, with rounded corners. The stays therefore arc per- 
pendicular to the plates. See Fig. 192. 

Fire-brick arches. These are used, as shown in Fig. 193, to 
force all the gases to circulate through the upper part of the fire- 



§318. 



ROLLING-STOCK. 



345 



box. Perfect combustion requires that all the carbon shall be 
turned into carbon dioxide, and this is facilitated by the 
forced circulation. 





,^ ooooo 

:;^OOOOCvOOOO ^^ 
;U. OOOOOOOOOOO^/i 

i^y-^r 'O o o O O O O O O O O O H 

ii'_i , o o o o o o c o oo o op 
-a-;^ 00000:0 ooooo oc"^' 




1 _^ 00 00010 ooooo oc_a_I| 

^-^ O OQOOOOOOO QO^HL 




V V V V ^ V 



Water-tables. The same object is attained by using a water- 
table instead of a brick arch— as shown in Fig. 191. But it has 



346 



RAILROAD CONSTRUCTION. 



§319. 



the lurther advantages of giving additional heating-surface and 
avoiding the continual expense of maintaining the bricks. One 




Fig. 192. — "Belpaire" Fire-box. 
Half -section through AB. Half -section through CD. 

feature of the design is the use of a number of steam- jets 
which force air into the fire-box and assist the combustion. 





Fig. 1 93. — Fire-brick Arch. 



Fig. 194. — Wootten Fire-box. 



Area. Fire-boxes are usually limited in width to the prac- 
ticable width between the wheels — thus giving a net inside 
width of about 3 feet and a maximum length of 10 to 11 feet — 
this being about the maximum distance over which the firemen 
can properly control the fire. About 37 square feet is the 
maximum area obtainable except when the ^'Wootten" fire- 
box is used — illustrated in Fig. 194. Here the grate is raised 
above the driving-wheels and has (in the case shown) a width 
of 8' 0|". The fire-box area is over 76 square feet. Note that 
two furnace-doors are used. 

9. Coal consumption. No form of steam-boiler (except 
a boiler for a steam fire-engine) requires as rapid production 
of steam, considering the size of the boiler and fire-box, as a 



§ 319. ROLLIXG-STOCK. 347 

locomotive. The combustion of coal per square foot of grate 
per hour for stationary^ boilers averages about 15 to 25 lbs. and 
seldom exceeds that amount. An ordinary maximum for a 
locomotive is 125 lbs. of coal per square foot of grate-area per 
hour, and in some recent practice 220 lbs, have been used. Of 
course such excessive amounts are wasteful of coal, because 
a considerable percentage of the coal will be blown out of the 
smoke-stack unconsumed, the draft necessary for such rapid 
consumption being very great. The only justification of such 
rapid and wasteful coal consumption is the necessity for rapid 
production of steam. The best quality of coal is capable of 
evaporating about 14 lbs. of water per pound of coal, i.e.. change 
it from water at 212° to steam at 212°; the heat required to 
change water at ordinary- temperatures to steam at ordinary 
working pressure is (roughly) about 20% more. From 6 to 9 lbs. 
of water per pound of coal is the average perfoniiance of ordinary 
locomotives, the efficiency being less with the higher rates of 
combustion. Some careful tests of locomotive coal consump- 
tion gave the following figures: when the consumption of coal 
was 50 lbs. per square foot of grate-area per hour, the rate of 
evaporation was 8 lbs. of water per pound of coal. When the 
rate of coal consumption was raised to 180, the evaporation 
dropped to 5 lbs. of water per pound of coal. It has been 
demonstrated that the efficiency of the boiler is largely increased 
by an increased length of boiler-tubes. The actual consump- 
tion of coal per mile is of course an exceedingly variable quan- 
tity, depending on the size and type of the engine and also on 
the work it is doing — whether climbing a heavy grade with its 
maximtim train-load or running easily over a level or down 
grade. A test of a 50-ton engine, running without am^ train at 
about 20 to 25 miles per hour, showed an average consumption 
of 21 lbs. of coal per mile. Statistics of the Pennsylvania Rail 
road show a large increase (as might be expected, considering 
the growth in size of engines and weight of trains) in the aver- 
age number of pounds of coal burned per train-nnle — some of 
the figures being 55 lbs. in 1863, 72 lbs. in 1872, and nearly 
84 lbs. in 1883. Figures are published showing an average 
consumption of about 10 lbs. of coal per passenger-car mile, 
and 4 to 5 lbs. per freight-car mile. But these figures are always 
obtained by dividing the total constimption per train-mile by 
the number of cars, the coal due to the weight of the engine 



348 RAILROAD CONSTRUCTION. • § 320, 

being thrown in Wellington developed a rule, based on the 
actual performance of a very large number of passenger-trains, 
that the number of pounds of coal per mile = 21.1 + 6 74 times 
the number of passenger-cars The amount of coal assigned 
to the engine agrees remarkably with the test noted above 
For freight-trains the amount assigned to the engine should 
be much greater (since the engine is much heavier), and that 
assigned to the individual cars much less, although the great 
increase in freight-car weights in recent years has caused an 
increase in the coal required per car. 

320. Heating- surface. The rapid production of steam re- 
quires that the hot gases shall have a large heating-surface to 
which they can impart their heat From 50 to 75 square feet 
of heating-surface is usually designed for each square foot of 
grate-area. A more recently used rule is that there should be 
from 60 to 70 square feet of tube heating-surface per square 
foot of grate-area for bituminous coal 40 or 50 to 1 is more 
desirable for anthracite coal Almost the whole surface of 
the fire-box has w^ater behind it, and hence constitutes heating- 
surface. Although this surface forms but a small part of the 
total (nominally), it is really the most effective portion, since 
the difference of temperature of the gases of combustion and 
the water is here a maximum, and the flow of heat is therefore 
the most rapid. The heating-surface of the tubes varies from 
85 to 93% of the total, or about 7 to 15 times the heating-sur- 
face in the fire-box. Sometimes the heating-surface is as much 
as 2300 square feet, but usually it is less than 2000, even for 
engines w^hich must produce steam rapidly. 

Some of the most recent locomotives have greatly exceeded 
these figures One just constructed for the New York Central 
and Hudson Rivei Railroad has the following figures* heating- 
surface, 3500 sq. ft ; grate-area, 50 sq ft ; cylinders, 21'' X 26"; 
total weight, 176000 lbs : weight on drivers, 95000 lbs.; drivers, 
79'' diameter; wnth 85% of the boiler pressure, it developed 
an adhesion of 24700 lbs., which represented a factor of adhesion 

Another rule used by designers is that the engine should 
have 1 sq ft of heating-surface for each 50 of 60 lbs of weight, 
efficiency being indicated by a low weight. For the above 
engine the ratio is 53 



§ 321. ROLLING-STOCK. 349 

321. Loss of efnclency in steam pressure. The effective 
work done b^- the piston is never equal to the theoretical energy 
contained in the steam withdrawn from the boiler. This is due 
chiefly to the following causes: 

(a) The steam is ^'wire-drawn/' i.e., the pressure in the 
cylinder is seldom more than 85 to 90% of the boiler pressure. 
This is due largely to the fact that the steam -ports are so small 
that the steam cannot get into the cylinder fast enough to exert 
its full pressure. It is often purposely wire-drawn by partially 
closing the throttle, so that the steam may be used less rapidly. 

(b) Entrained water. Steam is always drawn from a dome 
placed over the boiler so that the steam shall be as far above 
the water-surface as possible, and shall be as dry as possible. 
In spite of this the steam is not perfectly dry and carries with 
it water at a temperature of, say, 361°, and pressure of 140 lbs 
per square inch. Vv'hen the pressure falls during the expan- 
sion and exhaust, this hot water turns into steam and absorbs 
the necessary heat from the hot cylinder-walls. This heat is 
then carried out by the exhaust and wasted. 

(c) The back pressure of the exhaust-steam, which depends 
on the form of the exhaust-passages, etc. This amounts to 
from 2 to 20% of the power developed. 

(d) Clearance-spaces. V/hen cutting off at full stroke this 
waste is considerable (7 to 9%), but when the steam is used 
expansively the steam in these clearance-spaces expands and 
so its power is not wholly lost. 

(c) Radiation. In spite of all possible care in jacketing the 
cylinders, some heat is lost by radiation. 

(/) Radiation into the exhaust-steam. This is somewhat 
analogous to (b). Steam enters the cylinder at a temperature 
of, say, 301°; the walls of the cylinder are much cooler, say 250°; 
some heat is used in raising the temperature of the cylinder- 
walls; some steam is vaporized in so doing; vrhen the exhaust 
is opened the temperature and pressure fall; the heat tem- 
porarily absorbed b}^ the C3dinder-walls is reabsorbed by the 
exhaust-steam, re-evaporating the vapor previously formed, 
and thus a certain portion of heat-energy goes through the 
cylinder v.dthout doing any useful A^ork. With an early cut-off 
the loss due to this cause is A^ery great. 

The sum of all these losses is exceedingh^ variable. They 
are usually less at lower speeds. The loss in initial pressure 



350 RAILROAD CONSTRUCTION. § 322. 

(the difference between boiler pressure and the cylinder pres- 
sure at the beginning of the stroke) is frequently over 20%, 
but this is not all a net loss With an early cut-off the average 
c\'linder pressure for the whole stroke is but a small part of 
the boiler pressure, yet the horse-power developed may be as 
great as, or greater than that developed at a lower speed, later 
cut-off, and higher average pressure 

322. Tractive power The work done by the two cylinders 
during a complete revolution of the drivers evident!}^ =area of 
pistons X average steam pressure X stroke X2X2. The resist- 
ance overcome evidently ^tractive force at circumference of 
drivers times distance traveled by drivers (which is the cir- 
cumference of the drivers) Therefore 

( area pistons X average steam pressure 

rr 4.- . < XstrokeX2x2, 

Tractive force = ) -. ,-. . 

C circumierence of drivers 

Dividing numerator and denominator by tt (3 1415), we have 

C (diam piston) ^ X average steam 

n^ J.' r < pressure X stroke /^o^^n 

Tractive force = ) ~ j-^-. , , (137) 

(. diameter 01 driver 

which is the usual rule Although the rule is generally stated 
in this form, there are several deductions In the first place 
the net effecti^^e area of the piston is less than the nominal on 
account of the area of the piston-rod. The ratio of the areas 
of the piston-rod and piston varies, but the effect of this reduc- 
tion is usually from 1.3 to 17% No allowance has been made 
for friction — of the piston, piston-rod, cross-head, and the 
various bearings This A^ould make a still further reduction 
of several per cent. Nevertheless the above simple rule is 
used, because, as will be shown, no great accuracy can be 
utilized. 

The tractive force is limited by the adhesion between the 
drivers and the rails, and this is a function of the weight on the 
drivers. Under the most favorable conditions this has been 
tested to amount to one-third the weight on the drivers, but 
such a ratio cannot be depended on Wellington used the 
ratio one-fourth The Baldwin Locomotive Works in their 
''Locomotive Data" give tables and diagrams based on i, j%, 



§ 322. ROLLING-STOCK. 351 

and I adhesion. As low a value as J or even | is occasionally 
used, but such a low rate of adhesion would only be found when 
the rails were abnormally slippery. In a well-designed loco- 
motive the tractive force, as computed above, and the tractive 
adhesion should be made about equal. The uncertainty in 
the coefficient of adhesion shows the futility of any refinement 
in the computation of the tractive force. 

It is only at very slow speeds that an engine can utilize all 
of its tractive force. When running at a high speed, the utmost 
horse-power that the engine can develop will only produce a 
draw-bar pull, which is but a small part of the possible tractive 
force. Power is the product of force times velocity. If the 
power is constant and the ^^locity increases, the force must 
decrease. This fact is well shown in the figures of some tests 
of a locomotive. The dimensions were as follows: cylinders, 
18"X24"; drivers, 68'^; weight on drivers, 60000 lbs. ; heating- 
surface, 1458 sq. ft.; grate-area, 17 sq. ft. During one test 
the average cylinder pressure was 83.3 lbs. (boiler pressure, 
145); 14-inch cut-off and throttle | open). By the above 
formula (137), 

^ ^. „ 18^X83.3X24 ^^^^ ,, 
Tractive force = ^- =9525 lbs. 

Do 

At i adhesion the tractive force was 15000 lbs; even at i ad- 
hesion, it would be 12000 lbs. This shows that at the speed 
of :his test (26.3 in. per hour) scarcely more than J of the trac- 
tive power was utilized. A still more marked case, shown by 
another test with the same engine, taken when the speed was 
53.4 miles per hour, indicated an average cylinder pressure of 
37.2 lbs., the throttle being .^ open and the valves cutting off 
at 8". In this case the tractive power, computed as before, 
equals 4254 lbs., about -^:^ of the weight on the drivers and 
about i of the tractive force which is possible at slow speeds. 
In the first case, the tractive power (9525) times the speed in 
feet per second (38.57) divided by 550 gives the indicated horse- 
power, 668. In the second case, although the tractive forces 
developed was so much less, the speed was much greater and 
the horse-power was about the same, 606. 

The above figures illustrate some of the foregoing statements 
regarding loss of efficiency. In both cases the steam was wire- 
drawn. The boiler pressure was 145 lbs., but wheu the throttle 



352 RAILROAD CONSTRUCTION. § 322. 

was only J open and the steam was cut-off at 14" (24" stroke) 
the average steam pressure in the cylinder was reduced to 
83.3 lbs. With the throttle but I open and* the valves cutting 
off at 8" (i of the stroke), the average pressure was cut down 
to 37.2 lbs. — about J of the boilt3r pressure. Note that the heat- 
ing-surface per square foot of grate-area (1458-^ 17 ==-^86) is 
very large (see § 320). Note also that the horse-power developed 
divided bv the grate-area (17) gives 39 and 30 H.P. per square 
foot of grate-area. This is exceptionally large — 25 or 30 being 
a more common figure. 

The maximum tractive power is required when a train is 
starting, and fortunately it is at low velocities that the maxi- 
mum tractive force can be developed. The motion of the 
piston is so slow that there is but little reduction of steam 
pressure, and the valves are generally placed to cut off at full 
stroke. For the above engine, with 145 lbs. boiler pressure, 

18^X145X24 

the absolute m.aximum of tractive force is — tt^t = 

68 

16581 lbs. Of course, this maximum would never be reached 
unless the boiler pressure were increased. A common rule is 
to consider that the average effective CAdinder pressure for slow 
speed and full stroke will be 80% of the boiler pressure. This 
would reduce the tractive force to the (nominal) value of 13265 
lbs., and the corresponding cylinder pressure would be 116 lbs. 
.per square inch. With an effective cylinder pressure of about 
131 lbs. the tractive power is 15000 lbs., which is J of the total 
weight on the drivers. This illustrates the general rule, stated 
above, that the cylinders, drivers, and boiler pressure should 
be so proportioned that the maximum tractive force should 
about equal the maximum adhesion which could be obtained. 

As another numerical example, the dimensions of a recently 
constructed heavy consolidation engine are quoted. The cylin- 
ders are 24"X32"; diameter of drivers, 54"; total weight of 
engine and tender, 391400 lbs.; weight of engine, 250300 lbs.; 
weight on drivers, 225200 lbs.; capacit}^ of tender, 7500 gallons; 
the boiler has 406 tubes, 2|" in diameter and 15' long; fire- 
box, 132" X 40 J"; heating-surface of tubes, 3564 sq. ft.; of 
fire-box, 241 sq. ft. — total, 3805 sq. ft.; boiler pressure, 220 lbs. 
per square inch. Applying Eq. 132, we may compute 75093 
lbs. as the absolute maximum of tractive power. In fact this 
is an unattainable limit, for reasons before stated. The trac- 



§ 323. ROLLING-STOCK. 353 

tive force is given as 63000, which correspond? to an effective 
cvlinder pressure of about 185 lbs., about 84% of the boiler 
pressure. This tractive force is 28% of the weight on the 
drivers, a tractive ratio of 1 : 3.6. 

RUNNING GEAR. 

323. Types of running gear, (a) "American." This was 

Q/^ once the almost universal type for 
2-^ Li ^ A both passenger and freight service. 

It is still very conimonly used for passenger service, but it is 
not the best form for heavy freight work. 

(b) **Columbia." Four driv^ers, one pair of pilot-truck wheels 
and one pair of trailing wheels be- /^>^ ^ ^-^ 

hind the drivers. The low trailing ^-^ \ J ^^ — ^ -^ 

wheels permit a desirable enlargement of the fire-box. This 
is a recent type, used exclusively for passenger service, 

Q/^ (c) "Atlantic." Similar to 
L-Z Q Q -^ h except that the pilot -truck 

has four wheels instead of two. 

(d) "Mogul." These are used for both passenger and freight 
service, but are not ^vell 
adapted for either high speed 
or great tractive power. 

(e) "Ten-wheel." Similar to d except that the pilot-truck 

Q^-^ x^ has four wheels instead of 

V y V y 00 two. The use is similar to 

that of d. 

(f) "Consolidation." The present standard for freight ser- 
\4ce. It permits great trac- i^~*\ /^^ i^~N r \^ 

tive power without excessive x J — \^ — v_y — \^J lJ -A 

concentrated loads on the track. 

(g) Switching-engines. These have four or six (and excep- 
tionally even eight or ten) drivers and no truck-wheels. They 
are only adapted for slow speed when a maximum of tractive 
power is needed. Sometimes the water-tank and even a small 
fuel-box is loaded on. Since fuel is always near at hand for a 
yard -engine, the fuel-box need not be large. 

(h) "Double-enders." As explained in § 315, truck- wheels are 
needed in front of the drivers to guide them around curves. If 
an ordinary engine is run backward, the flanges of the rear 



o O O o - 



354 RAILROAD CONSTRUCTION. § 324. 

drivers will become badly worn, and if the speed is high, the 

danger of derailment is considerable. In suburban service, 

-^ /^^ /^^ -^ when the runs are short, it is 

sA — ^^^ ^^-^ — ^ preferable to run the engines 

forward and backw^ard, rather than turn them at each end of 
the route. Therefore a pilot-trutk is placed at each end. 

(i) " Miscellaneous types." Almost every conceivable com- 
bination of drivers and truck-wheels has been used. The 
^^ Mastodon" is similar to the ^'Consolidation" except that the 
pilot-truck has four wheels instead of two. The ^'Decapod" 
has ten driving-wheels. The ''Forney" (named after the in- 
ventor) has been very extensively used on elevated roads. The 
w^eight of the boiler and machinery is carried on four driving- 
wheels; the engine-frame is extended so as to include a small 
tank and fuel-box, the weight of which is chiefly supported by 
a truck of two or four wheels. They run best when running 
''backward," i.e., tender first. 

324. Equalizing-levers. The ideal condition of track, from 
the standpoint of smooth running of the rolling stock, is that 
the rails should always lie in a plane surface. While this con- 
dition is theoretically possible on tangents, it is unobtainable 
on curves, and especially on the approaches to curves when the 
outer rail is being raised. Even on tangents it is impossible 
to maintain a perfect surface, no matter how perfectly the 
track may have been laid. In consequence of this, the points 
of contact of the wheels of a locomotive, or even of a four- 
wheeled truck, will not ordinarily lie in one plane. The rougher 
and more defective the track, the worse the condition in this 
respect. Since the frame of a locomotive is practically rigid, 
if the frame rests on the driver-axles through the medium of 
springs at each axle-bearing, the compression of the springs 
(and hence the pressure of the drivers on the rail) will be varia- 
ble if the bearing-points of the drivers are not in one plane 
This variable pressure affects the tractive power and severely 
strains the frame. Applying the principle that a tripod will 
stand on an even surface, a mechanism is employed which 
virtually supports the locomotive on three points, of w^hich one 
is usually the center-bearing of the forward truck. On each 
side the pressure is so distributed among the drivers that even 
ii a driver rises or falls with reference to the others, the load 
carried by each driver is unaltered, and that side of the engine 



§324. 



ROLLING-STOCK. 



355 



rises or falls by one ?2th of the rise or fall of the single driver, 
where n represents the number of wheels. The principle in- 
volved is shown in an exaggerated form in Fig. 195. In the 
diagram, MX represents the normal position of the frame A\hen 
the wheels are on line. The frame is supported by the hanger^ 
at a, c, /, and h. ah, de, and gli are horizontal levers vibrating 
about the points H, K, and L, which are supported by the 
axles. While it is possible with such a system of levers to make 
MN assume a position not parallel with its natural position, 
yet, by an extension of the principle that a beam balance loaded 
with equal weights will always be horizontal, the effect of rais- 
ing or lowering a wheel will be to move il/iV parallel to itself. 




771--^::. 



Fig. 195. — Action of Equalizing-levers. 

It only remains to determine hoiu much is the motion of MN 
relative to the rise or drop of the wheel. 

The dotted lines represent the positions of the wheels and 
levers when one wheel drops into a depression. The wheel 
center drops from p to q, a distance m, L drops to L', a 
distance m (see Fig. 195, h) ; M drops to M ', an unknown dis- 
tance x; therefore aa^=x; hh' =x', cc' =^x\ dd! =^Zx = ee' \ f]' =x\ 
.\gg' = 5x; hh'=x', LU = h{gq' + h]i')=h{^x)=m\ .\ x = lm.\ 
i.e., MN drops, parallel to itself, l/n as much as the wheel 
drops, where n is the number of wheels. The resultant effect 
caused bv the simultaneous motion of two wheels with refer- 



356 RAILROAD CONSTRUCTION. § 324, 

ence to the third is evidently the algebraic sum of the effects 
of each wheel taken separately. 

The practical benefits of this device are therefore as follows : 

(a) When any driver reaches a rough place in the tracks a 
high place or a low place, the stress in all the varioxis hangers 
and levers is unchanged. 

(b) The motion of the frame (represented by the bar MN 
in Fig. 195) is but l/n of the motion of the wheel, and the jar 
and vibration caused by a roughness in the track is correspond- 
ingly reduced. 

The details of applying these principles are varied, but in 
general it is done as follows; 

(a) American and ten-wheeled types. Drivers on each side 
form a system. The center-bearing pilot-truck is the third 
point of support. The method is illustrated in Fig. 196. 

(b) Mogul and consolidation types. The front pair of drivers 
is connected with the two-wheeled pilot-truck (as illustrated 
in Fig. 197) to form one system. The remaining drivers on 
each side are each formed into a system 

The device of equalizers is an American invention. Until 
recently it has not been used on foreign locomotives. The 
necessity for its use becomes less as the track is maintained 
with greater perfection and is more free from sharp curves. 
A locomotive not equipped with this device would deteriorate 
very rapidly on the comparatively rough tracks which are 
usually found on light-traffic roads. It is still an open ques- 
tion to what extent the neglect of this device is responsible 
for the statistical fact that average freight-train loads on foreign 
trains are less in proportion to the weight on the drivers than 
is the case with American practice. The recent increasing use 
of this device on foreign heavy freight locomotives is perhaps 
an acknowledgment of this principle. 

325. Counterbalancing. At very high velocities th« cen- 
trifugal force developed by the weight of the rotating parts 
becomes a quantity which cannot be safely neglected. These 
rotating parts include the crank-pin, the crank-pioi boss, the 
side rod, and that part of the weight of the connecting-rod 
which may be considered as rotating about the center of the 
crank-driver. As a numerical illustration, a driving-wheel 
62" in diameter, running 60 miles per hour, will revolve 325 
times per minute. The weights are: 



§324. 



ROLLING-STOCK, 



357 






-— r^/' 



358 RAILROAD CONSTRUCTION. § 325. 

Crank-pin 110 lbs. 

boss 150 '' 

One-half side rod 240 " 

Back end of connecting-rod 190 ^' 

Total 690 lbs. 

If the stroke is 24", the radins of rotation is 12'', or 1 foot. Then 
Gv^ 690 X4;r2l 2x3252 



gr 32.2X1X602 



= 24821 lbs., 



which is half as much again as the weight on a driver, 16000 lbs. 
Therefore if no counterbalancing were used, the pressure be- 
tween the drivers and the rail would always be less (at any 
velocity) when the crank-pin was at its highest point. At a 
velocity of about 48 miles per hour the pressure would become 
zero, and at higher velocities the wheel would actually be 
thrown from the rail. As an additional objection, when the 
crank-pin was at the lowest point, the rail pressure would be 
increased (velocity 60 miles per hour) from 16000 lbs. to nearly 
41000 lbs., an objectionably high pressure. These injurious 
effects are neutralized by ^'counterbalancing." Since all of 
the above-mentioned weights can be considered as concen- 
trated at the center of the crank-pin, if a sufficient weight is so 
placed in the drivers that the center of gravity of the eccentric 
weight is diametrically opposite to the crank-pin, this centrifu- 
gal force can be Avholly balanced. This is done by filling up 
a portion of the space between the spokes. If the center of 
gravity of the counterbalancing weight is 20'' from the center, 
then, since the crank-pin radius is 12", the required weight 
would be 690Xi|=414 lbs. 

In addition to the effect of these revolving parts there is 
the effect of the sudden acceleration and retardation of the 
reciprocating parts. In the engine above considered the 
weights of these reciprocating parts will be: 

Front end of connecting-rod 150 lbs. 

Cross-head 174 " 

Piston and piston-rod 300 ' ' 

Total 624 lbs. 



§ 325. ROLLING-STOCK. 359 

Assume as before that the reciprocating parts may be con- 
sidered as concentrated at one point, the point P of the dia- 
gram in Fig. 198. Since the motion of P is horizontal only, 



Fig. 198. — Action of Cox nterbalance. 

the force required to overcome its inertia at any point will 
exactly equal the horizontal coynponent of the force required 
to overcome the inertia of an equal weight at S, revolving in 
a circular path. Then evidently the horizontal component of 
the force required to keep W in the circular path will exactly 
balance the force required to overcome the inertia of P. Of 
course W=P. But a smaller weight IF', whose weight is 
inversely proportional to its radius of rotation, will evidently 
accomplish the same result. In the above numerical case, if 
the center of gravity of the counterw^eights is 20" from the 
center, the required w^eight to completeh^ counterbalance 
the reciprocating parts would be 624X^1 = 374.4 lbs. This 
counter^^eight need not be all placed on the driver carrying 
the main crank-pin, but can be (and is) distributed among all 
the drivers. Suppose it w^ere divided between the two drivers 
in the above case. At 60 miles per hour such a counterweight 
would produce an additional pressure of 11211 lbs. when the 
counterweight was down, or a lifting force of the same amount 
when the counterweight was up. Although this is not suffi- 
cient to lift the driver from the rail, it would produce an objec- 
tionabl}^ high pressure on the rail (over 27000 lbs.), thus inducing 
just w^hat it w^as desired to avoid on account of the eccentric 
rotating parts. Therefore a compromise must be made. Only 
a portion (one half to three fourths) of the weight of the recip- 
rocating parts is balanced. Since the effect of the rotating 
weights is to cause variable pressure on the rail, while the effect 
of the reciprocating parts is to cause a horizontal Avobuling or 
^' nosing'^ of the locomotive, it is impossible to balance bot^h. 
Enough counterweight is introduced to partially neutralize the 



360 RAILROAD CONSTRtrCTION. § 32j. 

effect of the reciprocating parts, still leaving some tendency 
to horizontal wobbling, while the counterweights which were 
introduced to reduce the w^obbling cause some variation of 
pressure. By using hollow piston-rods of steel, ribbed cross- 
heads, and connecting- and side-rods with an I section, the 
weight of the reciprocating parts may be greatly lessened with- 
out reducing their strength, and with a decrease in weight the 
effect of the unbalanced reciprocating parts and of the ^' excess 
balance'' (that used to balance the reciprocating parts) is 
largely reduced. 

Current practice is somewhat variable on three features: 

(a) The proportion of the weight of the connecting-rod which 
should be considered as revolving weight. 

(h) The proportion of the total reciprocating weight that 
should be balanced. 

(c) The distribution among the drivers of the counterweight 
to balance the reciprocating parts. 

An exact theoretical analysis of (a) shows that it is a func- 
tion of the weights and dimensions of the reciprocating parts. 
The weight which may be considered as revolving equals * 






in which r = radius of the crank, Z= length of connecting-rod, 
^= distance of center of gyration from wrist-pin, d = distance 
of center of gravity from WTist-pin, 1^1= weight of connecting- 
rod in pounds, and TT^2 = weight of piston, piston-rod, and cross- 
head in pounds; all dimensions in feet. An application of this 
formula will show that for the dimensions of usual practice, 
from 51 to 57% of the weight of the connecting-rod should be 
considered as revolving weight. 

The principal rules which have been formulated for counter- 
balancing may be stated as follows: 

1. Each wheel should be balanced correctly for the revolving 
parts connected with it. 

2. In addition, introduce counterbalance sufficient for 50% 
of the weight of the reciprocating parts for ordinary engines, 
increasing this to 75% when the reciprocating parts are exces- 
sively heavy (as in compound locomotives) or when the engine 

* R. A. Parke, in R. R. Gazette, Feb. 23, 1894. 



§ 326. 



ROLLING-STOCK. 



361 



is light and unable to withstand much lateral strain or when 
the wheel-base is short. 

3. Consider the weight of the connecting-rod as § revolving 
and J reciprocating when it is over 8 feet long; when shorter 
than 8 feet, consider -^^ of the weight as revolving and y\ as 
reciprocating. 

4. The part of the weight of the connecting-rod considered 
as revolving should be entirely balanced in the crank-driver 
wheel. 

5. The ^^ excess balance" should be divided equally among 
the drivers. 

6. Place the counterbalance as near the rim of the wheel 
as possible and also as near the outside 
of the wheel as possible in order that 
the center of gravity shall be as near 
as possible opposite the center of 
gravity of the rods, etc., which are all 
outside of even the plane of the face 
of the wheel. 

In Fig. 199 is shown a section of a 
locomotive driver with the cavities in 
the casting for the accommodation of 
the lead which is used for the couater- 
balance weight. Incidentally several 
other features and dimensions are shown 
in the illustration. 

326. Mutual relations of the boiler power, tractive power, 
and cylinder power for various types. The design of a locomo- 
tive includes tliree distinct features which are varied in their 
mutual relations according to the work which the engine is 
expected to do. 

(a) The boiler power. This is limited by the rate at which 
steam may be generated in a boiler of admissible size and weight. 
Engines which are designed to haul very fast trains which are 
comparatively light must be equipped ^^ ith very large grates and 
heating surfaces so that steam may be developed with great 
rapidity in order to keep up with the very rapid consumption. 
Engines for Aery heavy freight work are run at very much 
lower velocity and at a lower piston speed in spite of the fact 
that more strokes are required to cover a given distance and 
the demand on the boiler for rapid steam production is not 




Fig. 199. — Section of 
Locomotive -DRIVER. 



362 RAILROAD CONSTRUCTION. § 326 

as great as with high-speed passenger-engines. The capacity of 
a boiler to produce steam is therefore Hmited by the hmiting 
weight of the general type of engine required. Although im- 
provements may be and have been made in the design of fire- 
boxes so as to increase the steam-producing capacity without 
adding proportionate!}^ to the weight, yet there is a more or 
less definite limit to the boiler power of an engine of given 
weight. 

(b) The tractive power. This is a function of the weight on 
the drivers. The absolute limit of tractive adhesion between a 
steel-tired wheel and a steel rail is about one third of the pressure, 
but not more than one fourth of the weight on the drivers can 
be depended on for adhesion and wet rails will often reduce 
this to one fifth and even less. The tractive power is therefore 
absolutely limited by the practicable weight of the engine. In 
some designs, when the maximum tractive power is desired, not 
only is the entire weight of the boiler and running gear thrown 
on the drivers, but even the tank and fuel-box are loaded on. 
Such designs are generally employed in switching-engines (or 
on engines designed for use on abnormally heavy mountain 
grades) in which the maximum tractive power is required, but 
in which there is no great tax on the boiler for raj)id steam pro- 
duction (the speed being always very low), and the boiler and 
fire-box, which furnish the great bulk of the weight of an engine, 
are therefore comparatively light, and the requisite weight for 
traction must, therefore, be obtained b}' loading the drivers 
as much as possible. On the other hand, engines of the highest 
speed cannot possibly produce steam fast enough to maintain 
the required speed unless the load be cut down to a compara- 
tively small amount. The tractive power required for this 
comparatively small load will be but a small part of the weight 
of the engine, and therefore engines of this class have but a 
small proportion of their weight on the drivers; generally 
have but two driving-axles and sometimes but one. 

(c) Cylinder power. The running gear forms a mechanism 
which is simply a means of transforming the energy of the boiler 
into tractive force and its power is unlimited, ^^'ithin the prac- 
tical conditions of the problem. The power of the running 
gear depends on the steam pressure, on the area of the piston, 
on the diameter of the drivers, and on the ratio of crank-pin 
radius to wheel radius, or of stroke to driver diameter. It 



§ 326. 



ROLLING-STOCK. 



363 



is alwaj's possible to increase one or more of these elements 
by a relatively small increase of expenditure until the cylinders 
are able to make the drivers slip, assuming a sufficiently great 
resistance. Since the power of the engine is limited by the 
power of its weakest feature, and since the running gear is the 
most easily controlled feature, the power of the running gear 
(or the '' cylinder power") is always made somewhat excessive 
on all well-designed engines. It indicates a badly designed 
engine if it is stalled and unable to move its drivers, the steam 
pressure being normal. If it is attempted to use a freight- 
engine on fast passenger service, it will probably fail to attain 
the desired speed on account of the steam pressure falling. 
The tractive power and cyhnder power are superabundant, but 
the boiler cannot make steam as fast as it is needed for high 
speed, especially when the drivers are small. The practical 
result would be a comparatively low speed kept up with a forced 
fire. If it is attempted to use a high-speed passenger-engine 
on heavy freight service, the logical result is a slipping of the 
drivers until the load is reduced. The boiler power and cvlinder 
power are ample, but the w^eight on the drivers is so small that 
the tractive power is only sufficient to draw a comparatively 
small load. 

These relations between boiler, cylinder, and tractive powder 
are illustrated in the following comparative figures referring 
to a fast passenger-engine, a heavv freight-engine, and a switch- 
ing-engine. The weights of the passenger- and freight-engines 
are about the same, but the passenger-engine has onh^ 72% of 





Cylinders. 


Total 
Wght. 


Wt. on 
Driv'rs 


Heat- 
ing 
Sur- 
face, 

sq. ft. 


Grate 
area 
sq. ft. 


Steam 
Pres- 
sure in 
Boiler. 


Stroke. 


Kind. 


Diam. 
Driver. 


Fast passenger . 
Heavy freight . 
Switcher 


19'' X 24" 
20'' X 24" 
19" X 24" 


126700 
128700 
109000 


81500 
112600 
109000 


1831.8 
1498.3 
1498.0 


26.2 
31.5 

22.8 


180 
140 
160 





the tractive power of the freight. But the passenger-engine 
has 22% more heating-surface and can generate steam much 
faster: it makes less than two thirds as many strokes in cover- 
ing a given distance, but it runs at perhaps twice the speed 



364 RAILROAD CONSTRUCTION. § 326. 

and probably consumes steam much faster. The switch- 
engine is lighter in total weight, but the tractive power is nearly 
as great as the freight and much greater than the passenger- 
engine. While the heating-surfaces of the freight- and switch- 
ing-engines are practically identical, the grate area of the switcher 
is much less; its speed is always low and there is but little neces- 
sity for rapid steam development. 

While these figures show the general tendency for the relative 
proportions, and in this respect may be considered as typical, 
there are large variations. The recent enormous increase in 
the dead weight of passenger-trains has necessitated greater 
tractive power. This has been provided sometimes by using 
''Mogul'' and "ten-wheel" engines, which were originally 
designed for freight work. On the other hand, the demand 
for fast freight service, and the possibility of safel}^ operating 
such trains by the use of air-brakes, has required that heavy 
freight-engines shall be run at comparatively high speeds, and 
that requires the rapid production of steam, large grate areas, 
and heating surfaces. But in spite of these variations, the 
normal standard for passenger service is a four-driver engine 
carrying about two thirds of the Aveight of the engine on the 
drivers, which are very large; the normal standard for freight 
work is the "consolidation,'' with perhaps 90% of the weight 
on the drivers, which are small, but which must have the pony 
truck for such speed as it uses; and finally the normal standard 
for switching ser\dce has all the weight on the drivers and has 
comparatively low stea}n-producing capacity. 

327. Life of locomotives. The life of locomotives (as a 
whole) may be taken as about 800000 miles or about '22 to 24 
3^ears. While its life should be and is considered as the period 
between its construction and its final consignment to the scrap 
pile, parts of the locomotive may have been renewed more 
than once. The boiler and fire-box are especially subject to. 
renewal. The mileage life is much longer than formerl3\ This 
is due partly to better design and partly to the custom of 
drawing the fires less frequently and thereby avoiding some 
of the destructive strains caused by extreme alterations of 
heat and cold. Recent statistics give the average annual 
mileage on twenty-three leading roads to be 41000 miles. 



§ 328. ROLLING-STOCK. 365 

CARS. 

328. Capacity and size of cars. The capacity of freight -cars 
has been enormously increased of late years. About thirty 
years ago the usual live-load capacity for a box-car was about 
20000 lbs. In 1893 the standard box-car, gondola-cars, etc., 
of the Pennsylvania Railroad on exhibition at the Chicago 
Exposition, had a live-load capacity of 60000 lbs. and a dead 
weight of 30000 to 33000 lbs With a full load, the weight on 
each wheel is nearly 12000 lbs , which equals or exceeds the 
load usually placed on the drivers of ordinary locomotives. 
But now cars with a live-load capacity of 80000 lbs. are almost 
standard, 100000-lb. cars are very common, and even larger cars 
are made for special service. (See Fig. 200.) 

The limitation of the carrying capacity for some kinds of 
freight depends somewhat on the amount of live load that 
can be carried within given dimensions; for the cross-section of 
a car is limited to the extreme dimensions which may be safely 
run through the tunnels and through bridges as at present 
constructed, and the length is somewhat limited by the dif- 
ficulty of properly supporting an excessively hea^^ load, dis- 
tributed o^er an unusually long span, by a structure w^hich is 
subjected to excessive jar, concussion, compression, and ten- 
sion. The cross-sectional limit seems to have been scarcely 
reached yet, except, perhaps, in the case of furniture and carriage- 
cars, whose load per cubic foot is not great. The usual width 
of freight-cars is about 8 to 9 feet, w^hile parlor-cars and sleepers 
are generally 10 feet wide and sometimes 11 feet. The highest 
point of a train is usually the smoke-stack of the locomotive 
which is generally 14 feet above the rails and occasionally OA^er 
15 feet. A sleeping-car usually has the highest point of the 
car al^out 14 feet above the rails. Box-cars are usually about 
8 feet high (above the sills), with a total height of about 11' S'\ 
Refrigerator-cars are usually about 9' high and furniture-cars 
about 10' above the sills, the truck adding about 3' 3". The 
usual length is 34 feet, but 35 to 40 feet is not uncommon. 
Passenger-cars (day coaches) are usually 50 feet long, exclusive of 
the end platforms and weigh 45000 to 50000 lbs. Sixty pas- 

I sengers at 150 pounds apiece (a high average) will only add 
9000 lbs. to the weight. A parlor-car or sleeper is generalh^ 

j about 65 feet long exclusiA'e of the platforms, which add about 
a' 0''. The weight is anywhere from 60000 to 80000 lbs. 



366 RAILROAD CONSTRUCTION. § 329. 

The weight of the 25 or 30 passengers it may carn^ is hardly 
worth considering in comparison. 

329. Stresses to which car-frames are subjected. A car 
is structurally a truss, supported at points at some distance 
from the ends and subjected to transverse stress. There is, 
therefore, a change of flexure at two points between the trucks. 
Besides this stress the floor is subjected to compression when 
the cars are suddenly stopped and to tension when in ordinary 
motion, the tension being greater as the train resistance is 
greater and as the car is nearer the engine. The tension is 
sometimes relieved by means of continuous drawbars (see 
§ 331), but this affords no relief against impact during com- 
pression, which is reall}^ more destructive. The shocks, jars, 
and sudden strains to which the car-frames are subjected are 
very much harder on them than the mere static strains due to 
their maximum loads if the loads were quiescent. Consequently 
any calculations based on the static loads are practically value- 
less, except as a very rough guide, and previous experience 
must be relied on in designing car bodies. As evidence of the 
increasing demand for strength in car-frames, it has been re- 
cently observed that freight-cars, built some years ago and f 
built almost entirely of wood, are recpiring repairs of wooden 
parts which have been crushed in service, the wood being per- 
fectly sound as regards decay. 

330. The use of metal. The use of metal in car construction 
is very rapidly increasing,. The demand for greater strength 
in car-frames has grown until the wooden framing has become 
so heavy that it is found possible to make steel frames and 
trucks at a small additional cost, the steel frames being twice 
as strong and 3^et reducing the dead weight of the car about 
5000 lbs., a consideration of no small value, especially on roads 
having heavy grades. Another reason for the increasing use 
of metal is the great reduction in the price of rolled or pressed 
steel, while the cost of wood is possibly higher than before. 
The advocates of the use of steel advise steel floors, sides, etc. 
For box-cars a wooden floor has advantages. For ore and 
coal-cars an all-metal construction has advantages. (Fig. 201.) 
In Germany, where steel frames have been almost exclusively 
in use for many years, they have not 3^et been able to determine 
the normal age limit of such frames; none have yet ivorn out. 
The life is estimated at 50 to 80 year^. 




Fig. 200.— 100 ,000- lb. Box Car. 




Fig. 201. — Steel Coal Car. 




Fig. 202. — Wooden Box Car; Steel Frame. 
{To face page 366.) 



(I 01 



§ 331 



ROLLING-STOCK. 



367 



I Brake-beams are also best made of metal rather than wood, 
I' as ^^as formerly done. Metal brake-beams are generally nsed 
jon cars having air-brakes, as a wooden beam must be exces- 
|lsivety large and heavy in order to have sufficient rigidity. 
I Truck-frames (see Fig. 203), which were formerly made 

principally of wood, are now largely made of pressed steel. 

It makes a reduction in weight of about 3000 lbs. per car. 

The increased durability is still an uncertain quantity. 




Fig. 203. 

331. Draft-gear. These are of necessity made wath springs 
for all passenger- and freight-cars. Coal-jimmies are often 
ifastened together by links dropped over hooks, but the larger 
coal-cars require springs to absorb the shocks. There is a 
tonsiderable theoretical advantage in "continuous draft-gear,'' 
i.e. having a rod (or pair of rods) nmning continuously from 
end to end of the car so that there shall be no tensile stress on 
lithe car-body itself. But there are several objections in prac- 
tice, (a) The draft-rod, if there is but one, should be in the 
center line of the car, i.e. pass through the two truck-centers and 
I the king-pins, which is impracticable. This difficulty is some- 
times obviated in an objectionable way by nmning the draw- 
bar above the truck-center. A better method is to use a pair 
of rods, (b) The rod is of no value during compression, and 
it is the compression a car receives by minor collisions during 
switching which produces maximum injurj^ to the car-body 
and the draft-gear, (c) The rod is much more liable to injury 
and requires much more expensive repairs when injured. 



368 KAILROAD CONSTRUCTION. § 333. 

The older method is to bolt the beams holding the draft- 
gear to the under side of the ear-body. This form is objection- 
able owing to the fact that the push and pull, being transmitted 
through the car-body, act eccentrically, tend to loosen the 
draft-beams from the car-body, and in case of a violent col- 
lision have been known to actually buckle the car-body up- 
ward (the cars being ^'flats''). The fastening of the draft- 
gear to the car-body has been made more secure by using cast- 
iron keys, then still more so by running the beams back to 
the " transoms'' (the heavy cross-beams which support the car 
and transfer its weight to the trucks), then by making a double 
center sill extending through the length of the car. Another 
device is to run the draft-gear through the end sill and then 
the line of push and pull running through the car-frame instead 
of under it, the car-frame can furnish its maximum resistance. 

332. Gauge of wheels and form of wheel-tread. In Fig. 204 
is shown the standard adopted by the Master Car Builders' 
Association at their twentieth annual convention. Note the 
normal position of the gauge-line on the wheel-tread. In 
Fig. 114, p. 238, the relation of rail to wheel-tread is shown 
on a smaller scale. It should be noted that there is no definite 
position where the wheel-flange is absolutely ^' chock-a-block" 
against the rail. As the pressure increases the wheel mounts 
a little higher on the rail until a point is soon reached when the 
resistance is too great for it to mount still higher. By this 
means is avoided the shock of unyielding impact when the car 
sways from side to side. When the gauge between the inner 
faces of the wheels is greater or less than the limits given in 
the figure, the interchange rules of the Master Car Builders' 
Association authorize a road to refuse to accept a car from 
another road for transportation. At junction points of rail- 
roads inspectors are detailed to see that this rule (as well as 
many others) is complied with in respect to all cars offered 
for transfer. 

TRAIN-BRAKES. 

333. Introduction. Owing to the very general misappre- 
hension that exists regarding the nature and intensity of the 
action of brakes, a complete analysis of the problem is con- 
sidered justifiable. This misapprehension is illustrated by the 
common notion (and even practice) that the effectiveness of 



\ 



§333. 



ROLLING-STOCK. 



369 




Fig. 204. — M. C. B. Standard Wheel-tread and Axle. 



370 RAILROAD CONSTRUCTION. § 334. 

braking a car is proportional to the brake pressure, and there- 
fore a brakeman is frequentl}^ seen using a bar to obtain a 
greater leverage on the brake-wheel and using his utmost 
strength to obtain the maximum pull on the brake-chain while 
the car is skidding along with locked wheels. 

When a vehicle is moving on a track with a considerable 
velocity, the m^ass of the vehicle possesses kinetic energy of 
translation and the wheels possess kinetic energy of rotation. 
To stop the vehicle, this energy must be destroyed. The 
rotary kinetic energy will varj^ from about 4 to 8% of the 
kinetic energy of translation, according to the car loading 
(see § 347). On steam railroads brake action is obtained by 
pressing brake-shoes against car-wheel treads. As the brake- 
shoe pressure increases, the brake-shoes retard Avith increasing 
force the rotary action of the wheels. As long as the wheels 
do not slip or ^^skid" on the rails, the adhesion of the rails 
forces them to rotate with a circumferential velocity equal to 
the train velocity. The retarding action of the brake-shoe 
checks first the rotative kinetic energy (^^hich is small), and 
the remainder develops a tendency for the wheel to slip on the 
rail. Since the rotative kinetic energy is such a small per- 
centage of the total, it will hereafter be ignored, except as 
specifically stated, and it will be assum.ed for simplicity that 
the only work of the brakes is to overcome the kinetic energy 
of translation. The possible effect of grade in assisting or 
preventing retardation, and the effect of all other track resist- 
ances, is also ignored. The amount of the developed force 
which retards the train movement is limited to the possible 
adhesion or static friction between the wheel and the rail. 
When the friction between the brake-shoe and the w^heel ex- 
ceeds the adhesion between the wheel and the rail, the wheel 
skids, and then the friction between the wheel and the rail 
at once drops to a much less quantity. It must therefore be 
remembered at the outset that the retarding action of brake- 
shoes on wheels as a means of stopping a train is absolutely 
limited by the possible static friction between the braked 
wheels and the rails. 

334. Laws of friction as applied to this problem. Much of 
the misapprehension regarding this problem arises from a very 
conmion and widespread misstatement of the general laws of 
friction. It is frequently stated that friction is independent 



§ 334. ROLLING-STOCK. 371 

of the velocity and of the unit of pressure. The first of these 
so-called laws is not even approximately true. A ver}^ exhaus- 
tive series of tests were made by Capt. Douglas Galton on the 
Brighton Railway in England in 1878 and 1879, and by M. 
George Marie on the Paris and L3^ons Railway in 1879, with 
trains w^hich were specially fitted with train-brakes and with 
dynagraphs of various kinds to measure the action of the 
brakes. Experience proved that variations in the condition of 
the rails (wet or dry), and numerous irregularities incident to 
measuring the forces acting on a heavy body moving with a 
high velocity, were such as to give somewhat discordant re- 
sults, even when the conditions were made as nearly identical 
as possible. But the tests were carried so far and so persist- 
ently that the general laws stated below were demonstrated 
beyond question, and even the numerical constants were deter- 
mined as closely as they may be practically utilized. These 
laws may be briefly stated as foUow^s: 

(a) The coefficient of friction between cast-iron brake-blocks 
and steel tires is about .3 when the wheels are "just mov- 
ing"; it drops to about .16 when the velocity is about 30 miles 
per hour, and is less than .10 when the velocity is 60 miles per 
hour. These figures fluctuate considerably T^'ith the condition 
of the rails, wet or dry. 

(b) The coefficient of friction is greatest when the brakes 
are first applied; it then reduces very rapidly, decreasing 
nearly one third after the brakes have been applied 10 seconds, 
and dropping to nearly one half in the course cf 20 seconds. 
Although the general truth of this law was established beyond 
question, the tests to demonstrate the laAv of the variation of 
friction A^ith time of application were too few to determine 
accurately the numerical constants. 

(c) The friction of skidded wheels on rails is always very 
much less than the adhesion when the wheel is rolling on the 
rail — sometimes less than one third as much. 

(d) An anal3^sis of the tests all pointed to a law that the 
friction developed does not increase as rapidly as the intensitij 
of pressure increases, but this may hardly be considered as 
an established law. 

{e) The adhesion between the wheel and the rail appears to 
be independent of velocit3\ The adhesion here means the force 
tjiat must be developed before the wheel will slip on the rail. 



372 RAILROAD CONSTRUCTION. § 33d. 

The practical effect of these laws is shown by the following 
observed phenomena: 

(a) When the brakes are first applied (the velocity being 
very high), a brake pressure far in excess of the weight on the 
wheel (even three or four times as much) may be applied with- 
out skidding the wheel. This is partly due to the fact that 
the wheel has a very high rotative kinetic energy (which varies 
as the square of the velocity, and which must be overcome 
first), but it is chiefly due to the fact that the coefficient of 
friction at the higher velocity is very small (at 60 miles per 
hour it is about .07), while the adhesion between the wheel and 
the rail is independent of the velocity. 

(b) As the velocity decreases the brake pressure must be 
decreased or the wheels will skid. Although the friction de- 
creases with the time required to stop and increases with the 
reduction of speed, and these two effects tend to neutralize 
each other, yet unless the stop is very slow, the increase in 
friction due to reduction of speed is much greater than the 
decrease due to time, and therefore the brake pressure must 
not be greater than the weight on the wheel, unless momentarily 
while the speed is still very high. 

(c) The adhesion between wheels and rails varies from .20 
to .25 and over when the rail is dry. When wet and slippery 
it may fall to .18 or even .15. The use of sand T\'ill always 
raise it above .20, and on a dry rail, when the sand is not blo\^n 
away by wind, it may raise it to .35 or even .40. 

(d) Experiments were made with an automatic valve by 
which the brake-shoe pressure against the Avheel should be 
reduced as the friction increased, but since (1) the essential 
requirement is that the friction produced by the brake-shoes 
shall not exceed the adhesion between rail and wheel, and 
since (2) the rail-wheel adhesion is a very variable quantity, 
depending on whether the rail is wet or dry, it has been found 
impracticable to use such a valve, and that the best plan is to 
leave it to the engineer to vary the pressure, if necessary, by the 
use of the brake- valve. 

MECHANISM OF BRAKES. 

335. Hand-brakes. The old style of brakes consists of brake- 
shoes of some type which are pressed against the wheel-treads 



§ 335. 



ROLLING-STOCK. 



373 



by means of a brake-beam, which is operated by means of a 
hand-windlass and chain operating a set of levers. It is desir- 
able that brakes shall not be set so tightly that the wheels 
sh^ll be locked, and then slide over the track, producing 
iiat places on them, which are very destructive to the 
rolling-stock and track afterward, on account of the impact 
occasioned at each revolution. With air-brakes the maximum 
pressure of the brake-shoes can be quite carefully regulated, 
and they are so designed that the maximum pressure exerted 
by any pair of brake-shoes on the w^heels of any axle shall not 
exceed a certain per cent, of the weight carried by that axle 
w^hen the car is empty, 90% being the figure usually adopted 
for passenger-cars and 70% for freight-cars. Consider the 
case of a freight-car of 100000 lbs. capacity, weighing 33100 lbs., 
or 8275 lbs. on an axle, and equipped with a hand-brake which 
operates the levers and brake-beams, which are sketched in 
Fig. 205. The dead w^eight on an axle is 8275 lbs.; 70% of 




-&5792 



Fig. 205. — Sketch op Mechanism of Hand-brake. 



this is 5792 lbs., which is the maximum allowable pressure 
per brake-beam, or 2896 lbs. per brake-shoe. With the dimen- 
sions shown, such a pressure will be produced by a pull of about 
1158 lbs, on the brake-chain. The power gained by the brake- 
wheel is not equal to the ratio of the brake-wheel diameter 
to the diameter of the shaft, about which the brake-chain 
winds, which is about 10 to 1|. The ratio of the circumfer- 
ence of the brake-wheel to the length of chain wound up by 
one complete turn would be a closer figure. The loss of efli- 



374 RAILROAD CONSTRUCTION. § 336. 

ciency in such a clumsy mechanism also reduces the effective 
ratio. Assuming the effective ratio as 6:1 it would require a 
pull of 193 lbs. at the circumference of the brake-wheel to 
exert 1158 lbs. pull on the brake-chain, or 5792 lbs. pressure 
on the wheels at B, and even this will not lock the wheels whenv 
the car is empt}^. much less when it is loaded. Note that the 
pressures at .4 and B are unequal. This is somewhat objec- 
tionable, but it is unavoidable with this simple form of brake- 
beam. More complicated forms to avoid this are sometimes 
used. Hand-brakes are, of course, cheapest in first cost, and 
even with the best of automatic brakes, additional mechanism 
to operate the brakes by hand in an emergency is always pro- 
vided, but their slow operation when a quick stop is desired 
makes it exceedingly dangerous to attempt to run a train at 
high speed unless some automatic brake directly under the 
control of the engineer is at hand. The great increase in the 
average velocity of trains during recent 3^ears has only been 
rendered possible by the invention of automatic brakes. 

336. "Straight'' air-brakes. The essential constructive fea- 
tures of this form of brake are (1) an air-pump on the engine, 
operated by steam, which compresses air into a reservoir on 
the engine; (2) a ''brake-pipe'' running from the reservoir 
to the rear of the engine and pipes running under each car, 
the pipes having flexible connections at the ends of the cars 
and engine; (3) a cylinder and piston under each car which 
operates the brakes by a system of levers, the cylinder being 
connected to the brake-pipe. The reservoir on the engine 
holds compressed air at about 45 lbs. pressure. To operate the 
brakes, a valve on the engine is opened which allows the com- 
pressed air to flow from the reservoir through the brake-pipe 
to each cylinder, moving the piston, which thereby moves the 
levers and applies the brakes. The defects of this system are 
many: (1) With a long train, considerable time is required for 
the air to flow from the reservoir on the engine to the rear cars, 
and for an emergency-stop even this delay would often be 
fatal; (2) if the train breaks in two, the rear portion is not • 
provided with power for operating the brakes, and a dangerous 
collision w^ould often be the result; (3) if an air-pipe coupling 
bursts under any car, the whole system becomes absolutely 
helpless, and as such a thing might happen during some emer- 
gency, the accident would then be especially fatal. 



§ 337. ROLLING-STOCK. 375 

This form of brake has almost, if not entirely, passed out of 
use. It is here briefly described in order to show the logical 
development of the form which is novr in almost universal use, 
the automatic. 

337. Automatic air-brakes. The above defects have been 
overcome by a method which may be briefly stated as follows: 
A reservoir for compressed air is placed under each car and the 
tender; whenever the pressure in these reservoirs is reduced 
for any reason, it is automatically replenished from the main 
reservoir on the engine; whenever the pressure in the brake- 
pipe is reduced for any cause (opening a valve at any point of 
its length, parting of the train, or bursting of a pipe or coupler), 
valves are automatically moved under each car to operate the 
piston and put on the brakes. All the brakes on the train are 
thus applied almost simultaneously. If the train breaks in two, 
both sections will at once have all the brakes applied automati- 
cally ; if a coupling or pipe bursts, the brakes are at once applied 
and attention is thereby attracted to the defect; if an emer- 
gency should arise, such that the conductor desires to stop 
the train instantly without even taking time to signal to the 
engineer, he can do so by opening a valve placed on each car, 
which admits air to the train-pipe, which will set the brakes 
on the whole train, and the engineer, being able to discover 
instantly what had occurred, would shut off steam and do 
whatever else was necessary to stop the train as quickly as pos- 
sible. The most important and essential detail of this S3^stem 
is the ^^ automatic triple valve" placed under each car. Quot- 
ing from the Westinghouse Air-brake Company's Instruction 
Book, ^'A moderate reduction of air pressure in the train-pipe 
causes the greater pressure remaining stored in the auxiliary 
reservoir to force the piston of the triple valve and its slide- 
valve to a position which will allow the air in the auxiliary 
reservoir to pass directly into the brake-cylinder and apply the 
brake. A sudden or violent reduction of the air in the train - 
pipe produces the same effect, and in addition causes supple- 
mental valves in the triple valve to be opened, permitting the 
pressure from the train-pipe to also enter the brake-cylinder, 
augmenting the pressure derived from the auxiliary reservoir 
] about 20%, producing practically instantaneous action of the 
brakes to their highest efficiency throughout the entire train. 
When the pressure in the brake-pipe is again restored to an 



376 RAILROAD CONSTRUCTION. § 338. 

amount in excess of that remaining in the auxihary reservoir, 
the piston- and slide-valves are forced in the opposite direction 
to their normal position, opening communication from the train- 
pipe to the auxiliary reservoir, and permitting the air in the 
brake-cylinder to escape to the atmosphere, thus releasing the 
brakes. If the engineer wishes to apply the brake, he moves 
the handle of the engineer's brake-valve to the right, which 
first closes a port, retaining the pressure in the main reservoir, 
and then permits a portion of the air in the train-pipe to escape. 
To release the brakes, he moves the handle to the extreme 
left, which allows the air in the main reservoir to flow freely 
into the brake-pipe, restoring the pressure therein." 

338. Tests to measure the efficiency of brakes. Let v repre- 
sent the velocity of a train in feet per second; W, its weight; 
F, the retarding force due to the biakes; <i, the distance in feet 
required to make a stop; and g, the acceleration of gravity 
(32.16 feet per square second); then the kinetic energy pos- 
sessed by the train (disregarding for the present the rotative 

kinetic energy of the wheels) = - — . The work done in stop- 

ping the train =Fc?. .'. Fcl = ~^. The ratio of the retarding 

force to the weight, 

F v^ v^ 

In order to compare tests made under varying conditions, the 
ratio F-^W should be corrected for the effect of grade ( + or — ), 
if any, and also for the proportion of the weight of the train 
which is on braked wheels. For example, a train weighed 
146076 lbs., the proportion on braked wheels was 67%, speed 
60 feet per second, length of stop 450 feet, track level. Sub- 
stituting these values in the above formula, we find (F^W) 
= .124. This value is really unduly favorable, since the ordi- 
nary track resistance helps to stop the train. This has a value 
of from 6 to 20 lbs. per ton, averaging say 10 lbs. per ton dur- 
ing the stop, or .005 of the weight. Since the effect of this is 
small and is nearly constant for all trains, it may be ignored 
in comparative tests. The grade in this case was level, and 
therefore grade had no effect. But since only 67% of the 
weight was on braked wheels, the ratio, on the basis of all the 



i 



§ 339. ROLLING-STOCK. 377 

wheels braked, or of the weight reduced to that actually on the 
braked wheels, is 0.124^.07 = 0.185. This was called a ''good" 
stop, although as high a ratio as 0.200 has been obtained. 

339. Brake-shoes. Brake-shoes were formerly made of 
wrought iron, but when it was discovered that cast-iron shoes 
would answer the purpose, the use of v\TOught-iron shoes was 
abandoned, since the cast-iron shoes are so much cheaper. A 
cheap practice is to form the brake-shoe and its head in one 
piece, which is cheaper in first cost, but when the wearing-sur- 
face is too far gone for further use, the whole casting must be 
renewed. The ''Christie" shoe, adopted by the Master Car 
Builders' Association as standard, has a separate shoe which 
is fastened to the head by means of a wrought-iron key. The 
shoe is beveled Y' in a width of 3|" to fit the coned wheel. 
This is a greater bevel than the standard coning of a car-wiieel. 
It is perhaps done to allow for some bending of the brake- 
beam and also so that the maximum pressure (and wear) should 
come on the outside of the tread, rather than next to the flange, 
where it might tend to produce sharp flanges. By concen- 
trating the brake-shoe wear on the outer side of the tread, the 
wear on the tread is more nearly equalized, since the rail wT.ars 
the wheel-tread chiefly near the flange. This same idea is 
developed still further in the "flange-shoes," which have a 
curved form to fit the wheel-flange and which bear on the 
wheel on the flange and on the outside of the tread. It is 
claimed that b}^ this means the standard form of the tread is 
better preserved than when the wear is entirely on the tread. 
The Congdon brake-shoe is one of a type in ^hich TSTOught- 
iron pieces are inserted in the face of a cast-iron shoe. It is 
claimed that these increase the life of the shoe. 



CHAPTER XVI. 

TRAIN RESISTANCE. 

340. Classification of the various forms. The various resist- 
ances which must be overcome by the power of the locomotive 
may be classified as follows : 

(a) Resistances internal to the locomotive, w^hich include fric- 
tion of the valve-gear, piston- and connecting-rods, journal 
friction of the drivers; also all the loss due to radiation, con- 
densation, friction of the steam in the passages, etc. In short, 
these resistances are the sum-total of the losses by which thd 
power at the circumference of the drivers is less than the power 
developed by the boiler. 

(b) Velocity resistances, which include the atmospheric resist- 
ances on the ends and sides; oscillation and concussion resist- 
ances, due to uneven track, etc. 

(c) Wheel resistances, which include the rolling friction be- 
tween the vv'heels and the rails of all the wheels (including the 
drivers) ; also the journal friction of all the axles, except those 
of the drivers. 

(c?) Grade and curve resistances, which include those resist- 
ances which are due to grade and to curves, and which are not 
found on a straight and level track. 

(e) Brake resistances. As shown later, brakes consume 
power and to the extent of their use increase the energ}^ to 
be developed by the locomotive. 

(/) Inertia resistances. The resistance due to inertia is not 
generally considered as a train resistance because the energy 
which is stored up in the train as kinetic energy may be util- 
ized in overcoming future resistances. But in a discussion 
of the demands on the tractive power of the engine, one of the 
chief items is the energ}^ required to rapidly give to a starting 
train its normal velocity. This is especially true of suburban 
trains^ which must acquire speed very quickly in order that 

378 



i 



§ 341. TRAIN RESISTANCE. 379 

their general average speed between termini may be even reason- 
ably fast. 

341. Resistances internal to the locomotive. These are re- 
sistances which do not tax the adhesion of the drivers to the 
rails, and hence are frequently considered as not being a part 
of the train resistance properly so cLilled. If the engine were 
considered as lifted from the rails and made to drive a belt 
placed around the drivers, then all the power that reached the 
belt would be the power that is ordinarily available for adhe- 
sion, while the remainder would be that consumed internally 
by the engine. The power developed by an engine may be 
obtained by taking indicator diagrams which show the actual 
steam pressure in a cylinder at any part of a stroke. From 
such a diagram the average steam pressure is easily obtained, 
and this average pressure, multiplied by the length of the stroke 
and by the net area of the piston, gives the energy developed 
by one half-stroke of one piston. Four times this product 
divided by 550 times the time in seconds required for one stroke 
gives the 'indicated horse-power" Even this calculation 
gives merely the power behind the piston, which is several per 
cent, greater than the power which reaches the circumference 
of the drivers, owing to the friction of the piston, piston-rod, 
cross-head, connecting-rod bearings, and driving-wheel jour- 
nals. (See § 322, Chapter XV.) By measuring the amount 
of water used and turned into steam, and by noting the boiler 
pressure, the energy possessed b}^ the steam used is readily 
computed. The indicator diagrams will show the amount of 
steam that has been effective in producing power at the cylin- 
ders. The steam accounted for by the diagrams will ordinarily 
amount to 80 or 85% of the steam developed by the boiler, 
and the other 15 or 20% represents the loss of energy due to 
radiation, condensation, etc. From actual t^sts it has been 
found that the power consumed by an engine running light is 
about 11%, of that required by the engine when working hard 
in express freight service. But since the engine resistances 
(friction, etc.) are increased when it is pulling a load, it was 
estimated, after allowing for this fact, that about 15 or 16% 
of the power developed by the pistons was consumed by the 
engine, leaving about 84 to 85% for the train. 

342. Velocity resistances, (a) Aimosphcric, This consists of 
the head and tail resistances and the side resistance. The head 



380 RAILROAD CONSTRUCTION. § 343. 

and tail resistances are neaily constant for all trains of given 
velocity, varying but slightly with the van'ing cross-sections 
of engines and cars. The side resistance varies with the length 
of the train and the character of the cars, box-^cars or flats, etc. 
Vestibuling cars has a considerable effect in reducing this side 
i*esistance by preventing much of the eddying of air-currents 
between the cars, although this is one of the least of the ad 
vantages of vestibuling. Atmospheric resistance is generally 
assumed to vary as the square of the velocity, and although 
this may be nearh^ true, it has been experimentally demon 
st rated to be at least inaccurate. The head resistance is gen 
erally assumed to vaty as the area of the cross-section, but this 
has been definitely demonstrated to be very far from true. A 
freight-train composed partly of flat-cars and partly of box- 
cars will encounter considerably more atmospheric resistance 
than one made exclusively of either kind, other things being 
equal. The definite information on this subject is very unsat- 
isfactory,, but this is possibly due to the fact that it is of little 
practical importance to know just how much such resistance 
amounts to. 

(b) Oscillatory and concussive. These resistances are con- 
sidered to vary as the square of the velocity. Probably this 
is nearly, if not quite, correct on the general principle that such 
resistances are a succession of impacts and the force of impacts 
varies as the square of the velocity. These impacts are due to 
the defects of the track, and even though it were possible to 
make a precise determination of the amount of this resistance 
in any particular case, the value obtained would only be true 
for that particular piece of track and for the particular degree 
of excellence or defect which the track then possessed. The 
general improvement of track maintenance during late years 
has had a large influence in increasing the possible train-load 
by decreasing the train resistance. The expenditure of money 
to improve track will give a road a large advantage over a 
competing road with a poorer track, b}^ reducing train resist- 
ance, and thus reducing the cost of handling traffic. 

343. Wheel resistances, (a)' Rolling friction of the wheels. 
To determine experimentally the rolhng friction of w^hcels, 
apart from all journal friction, is a very difficult matter and 
has never been satisfactorily accomplished. Theory as well 
as practice shows that the higher and the more perfect the 



§ 343. TRAIN RESISTANCE. 381 

elasticity of the wheel and the surface, the less will be the roll- 
ing friction. But the determination, if made, would be of 
theoretical interest only. 

The combined effect of rolling friction and journal friction 
is determinable with comparative ease. From the nature of 
the case no great reduction of the rolling friction by any device 
is possible. It is only a very insignificant part of the total 
train resistance. 

(h) Journal friction of the axles. This form of resistance has 
been studied quite extensively by means of the measurement 
of the force required to turn an axle in its bearings under 
various conditions of pressure, speed, extent of lubrication, 
and temperature. The following laws have been fairly well 
established: (1) The coefficient of friction increases as the pres- 
sure diminishes; (2) it is higher at very slow speeds, gradually 
diminishing to a minimum at a speed corresponding to a train 
velocity of about 10 miles per hour, then slowly increasing 
with the speed; it is very dependent on the perfection of the 
lubrication, it being reduced to one sixth or one tenth, when the 
axle is lubricated by a bath of oil rather than by a mxre pad 
or wad of waste on one side of the journal; (3) it is much lower 
at higher temperature, and vice versa. The practical effect of 
these laws is shown by the observed facts that (1) loaded cars 
have a less resistance per ton than unloaded cars, the figures 
being (for speeds of about 10 to 20 miles per hour) : 

For passenger- and loaded freight-cars. . . 4 lbs. per ton 

' ' empty freight-cars 6 ' ' " ' ' 

'' street-cars 10'' '' '' 

' ' freight-trucks without load 14 ' ' " ' ' 

(2) When starting a train, the resistances are about 20 lbs. 
per ton, notwithstanding the fact that the velocity resistances 
are practically zero; at about 2 miles per hour it will drop to 
10 lbs. per ton and above 10 miles per hour it may drop to 
4 lbs. per ton if the cars are in good condition. (3) The re- 
sistance could probably be materially lowered if some practicable 
form of journal-box could be devised which would give a more 
perfect lubrication. (4) It is observed that freight-train loads 
must be cut down in Avinter by about 10 oi 15% of the loads 
that the same engine can haul over the same track in summer. 
This is due partly to the extra roughness and inelasticity of the 



o 



82 RAILROAD CONSTRUCTION. § 344. 



track in winter, and partly to increased radiation from the 
engine wasting some energy, but this will not account for all 
of the loss, and the effect, which is probably due largel}^ to the 
lower temperature of the journal-boxes, is very marked and 
costly. It has been suggested that a jacketing of the journal- 
boxes, which would prevent rapid radiation of heat and enable 
them to retain some of the heat developed by friction, would 
result in a saving amply repaying the cost of the device. 

Roller journals for cars have been frequently suggested, and 
experiments have been made with them. It is found that they 
are very effective at low velocities, greatly reducing the start- 
ing resistance, which is very high with the ordinary forms of 
journals. But the advantages disappear as the velocity in- 
creases. The advantages also decrease as the load is increased, 
so that with heavily loaded cars the gain is small. The excess 
of cost for construction and maintena-nce has been found to be 
more than the gain from power saved. 

344. Grade resistance. The amount of this may be com- 
puted with mathematical exactness. Assume that the ball 
or cylinder (see Fig. 206) is being drawn up the plane. If W 




Fig. 206. 

is the weight, N the normal pressure against the rail, and G 

the force required to hold it or to draw it up the plane with 

uniform velocity, the roUing resistances being considered zero 

or considered as provided for by other forces, then 

Wh 
G:W:h:d, or G==-^; 

but for all ordinary railroad grades, d = c to within a tenth of 

1%, i.e., G = = WX rate of grade. In order that the student 

(/ 

may appreciate the exact amoimt of this approximation the per- 
centage of slope distance to its horizontal projection is given in 
the following tabular form: 



§344. 



TRAIN RESISTANCE, 



383 



Grade in per cent. 


1 


2 


3 


4 


5 


Slope cUst.^jOO 

nor. dist. 


100.005 


100.020 


100.045 


100.080 


100.125 



Grade in per cent. 


6 


7 


8 


9 


10 


Slope dist. ^jQQ 

hor. dist. 


100.180 


100.245 


100.319 


100.404 


100.499 



This shows also the error on various grades of measuring with 
the tape on the ground rather than held horizontally. Since 
almost all railroad grades are less than 2% (where the error 
is but .02 of 1%), and anything in excess of 4% is unheard 
of for normal construction, the error in the approximation 
is generally too small for practical consideration. 

If the rate of grade is 1 : 100, G = W Xjio^ i.e., G = 20 lbs. 
per ton ; . * . for any per cent, of grade, G = (20 X per cent, of grade) 
pounds per ton. When moving up a grade this force G is to 
be overcome in addition to all the other resistances. AMien 
moving down a grade, the force G assists the motion and may 
be more than sufficient to move the train at its highest allow- 
able velocit3\ The force required to move a train on a level 
track at ordinary freight-train speeds (say 20 miles per hour) 
is about 7 lbs. per ton. A down grade of /^ of 1% will fur- 
nish the same power; therefore on a down grade of 0.35%, a 
freight-train would move indefinitely at about 20 miles per hour. 
If the grade were higher and the train were allowed to gain 
speed freely, the speed would increase until the resistance at 
that speed would equal W times the rate of grade, when the 
velocity would become uniform and remain so as long as the 
conditions were constant. If this speed was higher than a 
safe permissible speed, brakes must be applied and power 
wasted. The fact that one terminal of a road is considerably 
higher than the other does not necessarily imply that the 'extra 
power needed to overcome the difference of elcA^ation is a 
total waste of energy, especially if the maximum grades are 
so low that brakes will never need to be applied to reduce a 
dangerously high velocity, for although more povvcr must be 



384 RAILROAD CONSTRUCTION. § 346. 

used in ascending the grades, there is a considerable saving of 
power in descending the grades. The amount of this saAdng 
will be discussed more fully in Chapter XXIII. 

345. Curve resistanc3. Some of the principal laws will be 
here given v/ithout elaboration. A more detailed 4iscussion 
will be given in Chapter XXII. 

(a) While the total curve resistance increases as the degree 
of curve increases, the resistance per degree of curve is much 
greater for easy curves than for sharp curves; e.g., the resist- 
ance on the excessiA'eh^ sharp curves (radiums 90 feet) of the 
elevated roads of New York City is very much less per degree 
of curve than that on curves of 1° to 5°. (b) Curve resistance 
Increases with the velocity, (c) The total resistance on a 
curve depends on the central angle rather than on the radius; 
i.e., two curves of the same central angle but of different radius 
would cause about the same total curve resistance. This is 
p^artly explained by the fact that the longitudinal slipping will 
be the same in each case. (See § 311, Chapter XV.) In each 
case also the trucks must be twisted around and the wheels 
slipped laterally on the rails by the same amount J^. (See 
§ 312, Chapter XV.) 

346. Brake resistances. If a down grade is excessively steep 
so that brakes must be applied to prevent the train acquiring 
a dangerous velocity, the energy consumed is hopelessly lost 
without any compensation. When trains are required to make 
frequent stops and yet maintain a high average speed, consid- 
erable power is consumed by the application of brakes in stop- 
ping. All the energy which is thus turned into heat is hope- 
lessly lost, and in addition a very considerable amount of steam 
is drawn from the boiler to operate the air-brakes, which con- 
.=?ume the p)ower already developed. It can be easily demonstrated 
that engines drawing trains in suburban service, making fre- 
quent stops, and yet developing high speed between stops, will 
consume a very large proportion of the total power developed 
by the use of brakes. Note the double loss. The brakes con- 
sume power already developed and stored in the train as kinetic 
or potential energy, while the operation of the brakes requires 
additional steam power from the engine. 

347. Inertia resistance. The two forms of train resistance 
which under some circumstances are the greatest resistances 
to be overcome by the engine are the grade and inertia resist- 



§ 347. TRAIN RESISTANCE. 385 

ances, and fortunately both of these resistances may be com- 
puted with mathematical precision. The problem may be 
stated as follows: What constant force P (in addition to the 
forces required to overcome the various frictional resistances, 
etc.) will be required to impart to a body a velocity of v feet 
per second in a distance of s feet? The required number of 
foot-pounds of energy is evidently Ps. But this ■work imparts 

a kinetic energy which may be expressed by ~-^~' Equating 
these values, we have Ps=-^r—y or 

(138) 



2gs 



The force required to increase the velocity from ^^i to v^ may 

likewise be stated as P = ^ — ('^'2^~'^'i')- Substituting in the 

Zgs 

fornrjla the values TT^ = 2000 lbs. (one ton), gr = 32.16, and 5 = 

5280 feet (one mile), we have 

P = .00588(1^2' -O. 
Multiplying by (5280 -^ 3600)^ to change the unit of velocity 
to miles per hour, we have 

P = .01267(F22-Fi2)^ 

But this formula must be modified on account of the rotative 
kinetic energy which must be imparted to the wheels of the cars. 
The precise additional percentage depends on the particular 
design of the cars and their loading and also on the design of 
the locomotive. Consider as an example a box-car, 60000 lbs. 
capacity, weighing 33000 lbs. The wheels have a diameter 
of 36^' and their radius of gyration is about 13'''. Each wheel 
weighs 700 lbs. The rotative kinetic energy of each wheel is 
4877 ft.-lbs. when the velocity is 20 miles per hour, and for 
the eight wheels it is 39016 ft.-lbs. For greater precision 
(really needless^ we may add 192 ft.-lbs. as the rotative kinetic 
energy of the axles. When the car is fully loaded (weight 
93000 lbs.) the kinetic energy of translation is 1,244,340 ft.-lbs.; 
when empty (weight 33000 lbs.) the energy is 441540 ft.-lbs 
The rotative kinetic energy thus adds (for this particular 
car) 3.15% (when the car is loaded) and 8.9% (when the car 
is empty) to the kinetic energy of translation. The kinetic 



386 RAILROAD CONSTRUCTION. § 347. 

energy which is similarly added, owing to the rotation of the 
wheels and axles of the locomotive, might be similarly com- 
puted. For one type of locomotive it has been figured at about 
8%. The variations in design, and particularly the fluctua- 
tions of loading, render useless any great precision in these 
computations. For a train of ''empties" the figure would be 
high, probably 8 to 9%; for a fully loaded train it will not 
much exceed 3%. Wellington considered that 6% is a good 
average value to use (actually used 6.14% for ''ease of compu- 
tation '0, but considering (a) the increasing proportion of live 
load to dead load in modern car design, (h) the greater care 
now used to make up full train-loads, and (c) the fact that 
full train-loads are the critical loads, it would appear that 5% 
is a better average for the conditions of modern practice. Even 
this figure allows something for the higher percentage for the 
locomotive and something for a few empties in the train. There- 
fore, adding 5% to tjie coefficient in the above equation, we 
have the true equation 

P = .0133(T/V-1PV), (139) 

in which V2 and V^ are the higher and lower velocities respec- 
tively in miles per hour, and P is the force required per ton to 
impart that difference of velocity in a distance of one mile 
If more convenient, the formula may be used thus: 

P.=^^(TV-F,^), .... (140) 

o 

in which s is the distance in feet and Pj is the corresponding 
force. 

As a numerical illustration, the force required per ton to 
impart a kinetic energy due to a velocity of 20 miles per hour 
in a distance of 1000 feet will equal 

which is the equivalent (see § 344) of a 1.4% grade. Since the 
velocity enters the formula as F^ while the distance enters only 
in the first power, it follows that it will require four times the 
force to produce twice the velocity in the same distance, or 
that with the same force it will require four times the distance 
to attain twice the velocity. 



§ 348. TRAIN RESISTANCE. 387 

As another numerical illustration, if a train is to increase its 
speed from 15 miles per hour to 60 miles per hour in a distance 
of 2000 feet, the force required (in addition to all the other 
resistances) will be 

P 70.224(3600-225) , , ^ rtmu 

^1 = ^QQQ = 118.50 lbs. per ton. 

This is equivalent to a 5.9% grade and shows at once that it 
would be impossible unless there were a very heavy doT\Ti 
grade, or that the train was very light and the engine very 
powerful. 

348. Formulae for train resistance. These are generally given 
in one of the forms 

^=/^+^ («)l (141) 

in which R is the resistance per ton, / is a coefficient to be deter- 
mined, Y is the velocity in miles per hour, and c is a constant, 
also to be determined. These formulae disregard grade and 
curve resistances, inertia resistance, and the active resistance 
(or assistance) of wind, as distinct from mere atmospheric 
resistance. In short, they are supposed to give the resistance 
of a train moving at a uniform velocity over a straight and 
level track, there being no appreciable wind. Both formulae 
are empirical, since the resistances do not vary either directly 
or as the square of the velocity. Some resistances vary nearly 
as the square and some nearly as the first power. 

The quantity c represents the journal friction and rolling 
friction, and these are assumed to be constant, although care- 
ful tests of journal friction show that its variation with velocity 
is irregular (see § 343). This shows that such simple formulae 
must alwa^^s be inaccurate, but some formulae have been sug- 
gested, having either of these general forms, which agree very 
closely with the results of actual tests. 

(a) Searles^s formula, 

R = 4.82 + mo36V^+ -"QQ^^^' (^'^- of eng and tender)^ 

gross weight 01 tram 

in which i^= total resistance in pounds per ton and V is the 
velocity in miles per hour. This formula does not take account 
of any difference in the form of the train (whether box-cars or 



3S8 



RAILROAD CONSTRUCTION. 



§348 



fiats), which would have a great influence on the atmospheric 
resistance; neitlier does it take into account the relation of 
length to weight, or whether the cars are loaded or empty. Never- 
theless the results agree very closely with the determinations 
of actual train tests. If the resistance is computed according 
to this formula for a given class of engine (e.g., a heavy consoli- 
dation), and for various lengths of train, it is found that the 
resistance per ton of the gross weight of the train is much less 
when the train is long, and for a train of ordinary length the 
resistance hardl}^ mcreases as fast as the velocity until the 
velocity is great. 

According to this formula, a heavy consolidation engine draw- 
ing forty loaded freight-cars would have to overcome a resist- 
ance of about 8.2 lbs. per ton of the gross weight of the train 
at a velocity of 20 miles per hour. At a velocity of 10 miles 
per hour this resistance drops to about 5.7 lbs. per ton. And 
so the value of 8 lbs per ton, used by Wellington in his com- 
putations of the totaf powder of locomotive.s on grades, may be 
considered a safe figure, especially as the velocity at critical 
places may be assumed to be reduced as much as necessary. 

(b) Wellingto7i^s formul(£y 



R 



3.9 + .0065^2 + 


.571^2 
W ' 


3.9 + .00757=^ + 


.647= 
W ' 


6.0 + .0083F2 + 


.5772 
W ' 


6.0 + .0106y2 + 


TXT- 



. for loaded fiat-cars 



for loaded box-cars 



. . for empty fiat-cars 



for empty box-cars 



(143) 



Notice in these formulae the additional journal resistance (indi- 
cated by the constant term) for unloaded cars. The last term 
evidently indicates the atmospheric resistance. The middle 
term allows for the oscillatory resistances. Assuming the 
constant term and the coefficients to have been correctly de- 
termined, these formula) should be better than Searles's, since 
a choice of formula} can be made depending on the conditions. 
A train consisting partly of box-cars and partly of fiat-cars 
will have a higher resistance than is shown by any of the above 



§ 348. TRAIN RESISTANCE. 389 

formula? (and not a mean value), on account of the increased 
atmospheric resistance acting on the irregular form of the train. 

(c) Engineering News Jormulay 

i?=J + 2. (144) 

This formula belongs to class (a), Eq. 141. Its veiy simplicity 
makes it valuable for general use, but like the succeeding 
formula, it does not take account of variations in the form 
of the train, which have a very material influence on the train 
resistance. 

(d) D. K. Clark's formula, 

i?=-~ + S (for tons of 2240 lbs.) | 

1' i ,- . (145) 

7? = .00522^2 4- 7. 14 (for tons of 2000 lbs.) ) 

This is a very old formula, and is mentioned because all of 
Clark's formulae carr}^ much weight. But in this case the 
formula is quite defective. The constant term (7.14) repre- 
senting the journal and rolling friction is too large and thus 
the formula gives too large a resistance at low velocities; the 
coefficient of F2(. 00522) is less than in the other formula^, and 
so at very high velocities the figures would be less than those 
given by Searles's or Wellington's formulae, and less than the 
results of actual tests. J'or mean velocities the figures accord 
fairly well with those given by the other formulae and by actual 
tests. 

(e) Baldwin Locomotive Works formtda. The Baldwin Loco- 
motive Works have adopted a formula of their own as the result 
of the experience they have been able to accumulate. It is 
stated 

R=l + ^ (146) 

It is claimed that this formula agrees well wath actual tests, 
and in fact is based on the results of tests, but it evidently 
cannot allow for known variations in the length or character 
of the train. As a general formula for locomotives which are 
to pull any kind of a load, the formula is of more value for 
practical use than Searles's or Wellington's. 
• In Plate IX is shown graphically the resistance per ton of 



390 RAILROAD CONSTRUCTION. § 349. 

four trains according to these five formulae. For purposes 
of comparison of the formulae, the weight of engine and total 
weight of cars is made the same for the four trains. The resist- 
ance would therefore be the same by formulae (a), (c), (d), 
and (e). The differences would only appear when applying 
Wellington's formula. Assume the following as train-loads: 

(a) Engine, 64 tons ; loaded flat-cars, 648 tons 

(b) '' 64 " '' box- '' 648 '' 

(c) '' 64 ^^ empty flat- '' 648 '' 
{d) '' 64 " " box- " 648 '' 

When applying any of these fornmla^, due allowance must be 
made for grade and curve resistances, inertia resistances, and 
the possible retarding influence of a high wind must be con- 
sidered if it is a question of the power of a locomotive of given 
type to draw a given load up a given grade. 

349. Dynamometer tests. These are made by putting a 
'^dynamometer-car" between the engine and the cars to be 
tested. Suitable mechanism makes an automatic record of 
the force which is transmitted through the dynamometer at 
any instant, and also a record of the velocity at any instant. 
One of the practical difficulties is the accurate determination 
of the velocity at any instant when the velocity is fluctuating. 
When the velocity is decreasing, the kinetic energy of the train 
is being turned into work and the force transmitted through the 
dynamometer is less than the amount of the resistance which 
is actually being overcome. On the other hand, when the 
velocity is increasing, the dynamometer indicates a larger 
force than that required to overcome the resistances, but the 
excess force is being stored up in the train as kinetic energy. 
Grade has a similar effect, and the force indicated by the dy- 
namometer may be greater or less than that required at the 
given velocity on a level by the force which is derived from, 
or is turned into, potential energy. Therefore the resistance 
indicated b}^ the dynamometer of a train will not be that on a 
level track at imiform velocity, unless the track is actually 
level and the velocity really uniform. 

Dynamometer tests under other circumstances are there- 
fore of no value unless it is possible to determine the true 
velocity at any instant and its rate of change, and also to de- 
ter^iine the grade, Of course, the grade is easily found, aTd 




20 30 40 50 

Velocity in miles per hour 



^00 



I 




J 




(To lace page 390.) 



§ 350. 



TRAIN RESISTANCE. 



891 



allowance for an increase or decrease of kinetic or potential 
energy must therefore be made before it is possible to know 
how much force is being spent on the ordinary resistances. 

350. Gravity or " drop ** tests. Dynamometer tests require 
the use of a d3'namometer which is capable of measuring a 
force of several thousands of pounds, and which therefore 
cannot determine such values with a close percentage of accu- 
racy, especially if the force is small. A drop test utihzes the 
force of gra^dty^ w^hich may be measured with, mathematical 
accuracy. The general method is to select a stretch of track 
which has a uniform grade of about 0.7% and which is prefer- 
ably straight for two or three miles. On such a grade cars 
with running gear in good condition may be started by a push. 
The velocity will gradually increase until at some velocit}^ 
depending on the resistances encountered, the cars w^ill move 
uniformly. The only work recjuiring extreme care with this 
method is the determination of the velocity. If the velocity 
is fluctuating, as it is during the time when it is of the greatest 
importance to know the velocitv, it is not sufficient to deter- 
mine the time rquired to run some long measured distance, 
for the average velocity thus obtained would probably differ 

A B 




Fig. 207. — Loss in Velocity-head. 

considerably from the velocity at the beginning and end of that 
space. If the train consists of five cars or more, the velocity 
may be determined electrically (as described by Wellington 
in his ^'Economic Location," etc., p. 793 et seq.) from the 
automatic record made on a chronograph of the passage of the 
first wheel and the last, the chronograph also recording auto- 



392 RAILROAD CONSTRUCTION § 350. 

matically the ticks of a clock beating seconds. From this the 
exact time of the passage of the first and last wheels of the 
train of cars may be determined to the tenth or twentieth of a 
second. 

Velocity 'head. From theoretical mechanics we know that 
if a body descends through any path by the action of gravity, 
and is unaffected by friction, its velocity at any point in the 
direction of the path of motion is V=\/2gh. If the body is 
retarded by resistances, its velocity at any point will be less 
than this. If AM^ Fig. 207, represents any grade (exaggerated 
of course), then BJ^ CK, etc., represent the actual fall at an}^ 

point. Let BF represent the fall /ij, determined from Ai = ^ , 

if 

in which v^ is the actual observed velocity at /. Then JF= the 
velocity-head consumed by the resistances between A and J. 
If the train continues to K, the corresponding /^^ is CG; the 
remaining fall GK consists of GN^ (=JF, which is the velocity- 
head lost back of J) and A'/v, the velocity-head lost between J 
and K. At some velocity (Vn) on any grade, the velocity 
will not further increase and the line AFGHI will then be hori- 
zontal and at a distance (hn) = EI below A , . . E. The grade 
AM is the ^' grade of repose '^ for that velocity (Vn)] i.e., it is 
the grade that w^ould just permit the train to move indefinitely 
at the velocity Vn. The broken line AFGHI should really be 
a curve, and the grade of repose at any joint is the angle between 
AM and the tangent to that curve at the given point. The 
''grade of repose" by its definition gives the total resistance 
of the train at the particular velocity, or multiplying the grade 
of repose in per cent, by 20 gives the pounds per ton of resist- 
ance. Thus being able to determine the total resistance in 
pounds per ton at any velocity, the variation of total resistance 
with velocity may be determined, and then by varying the 
resistances, using different kinds of cars, empty and loaded, 
box-cars and flats, the resistances of the different kinds at 
various velocities may be determined. 



CHAPTER XVII 

COST OF RAILROADS. 

351. General considerations. Although there are many ele- 
ments in the cost of railroads which are roughly constant per 
mile of road, yet the published reports of the cost of railroads 
ditf(T very widely. The variation in the figures is due to several 
causes, (a) Economy requires that a road shall be operated 
and placed on an earning basis as soon as possible. Therefore 
the reported cost of a road during the first few years of its 
existence is somewhat less than that reported later. This is 
well illustrated when a long series of consecutive reports from 
an old-established road is available; nearly every year there 
will be shown an addition to the previous figures. And this 
is as it should be. The magnificent road-beds of some old 
roads cannot be the creation of a single season. It takes many 
years to produce such settled perfect structures, (h) A large 
part of the variation is due to a neglect to charge up "permanent 
improvements'' as additions to the cost of the road. For the 
first few years of the life of a road a great deal of work is done 
which is in reality a completion of the work of construction, 
and yet the cost of it is buried under the item "maintenance 
of way." For example, a long wooden trestle is replaced b}^ 
an earth embankment and a culvert. Since the original trestle 
is to be considered a temporary structure, the excess of the 
cost of the permanent structure over that of the temporary 
structure should evidently be considered as an addition to the 
cost of the road. But if the fiUing-in was done slowly, a few 
train-loads at a time, and the work scattered over man}^ years, 
the cost of operating the "nuid-train" has perhaps been buried 
under "maintenance" charges, (c) The reports from which 
many of the following figures were taken have not always 
analyzed the items of cost with the same detail as has been 
here attempted, and to that is probably due many of the ^^aria- 
tions and apparent discrepancies. 

393 



394 RAILROAD CONSTRUCTION. § 352. 

The various items of cost will be classified as follows: 

1. Preliminary financiering. 

2. Surveys and engineering expenses. 

3. Land and land damages. 

4. Clearing and grubbing. 

5. Earthwork. 

6. Bridges, trestles, and culverts 

7. Trackwork. 

8. Buildings and miscellaneous structures. 

9. Interest on construction. 
10. Telegraph line. 

352. Item I. PRELIMINARY FINANCIERING. The cost of this 
preliminary work is exceedingly variable. The work includes 
the clerical and legal work of organization, printing, engraving 
of stocks and bonds, and (sometimes the most expensive of all) 
the securing of a charter. This sometimes requires special 
legislative enactments, or may sometimes be secured from a 
State railroad commission. It has been estimated that about 
2% of the railway capital of Great Britain has been spent in 
Parliamentary expenses over the charters. These expenses 
are usually but a small percentage of the total cost of the enter- 
prise, but for important lines the gross cost is large, while the 
amount of money thus spent by organizations which have 
never succeeded in constructing their roads is sometimes enor- 
mous. 

353. Item 2. Surveys and engineering expenses. The 
comparison of a large number of itemized reports on the cost 
of construction shows that the cost of the ^'engineering" will 
average about 2% of the total cost of construction. This in- 
cludes the cost of surveys and the cost of laying out and super- 
intending the constructive work. The cost of mere surveying 
up to the time when construction actually commences has 
been variously quoted at $60, $75, and even $150 per mile. 
In exceptional cases the surveying for a few miles through some 
gorge might cost many times this amount, but $150 per mile 
may be considered an ordinary maximum for difficult country. 
On the other hand, much construction has been done over the 
western prairies after hasty surveys costing not much over 
$10 per mile. 

354. Item 3. LAND AND LAND DAMAGES. The cost of this 
item varies from the extreme, in which not only the land for 



§ 355. COST OF RAILROADS. 395 

right-of-way but also grants of public land adjoining the road 
are given to the corporation as a subsidy, to the other extreme, 
where the right-of-way can only be obtained at exorbitant 
prices. The width required is variable, depending on the 
width that may be needed for deep cuts or high fills, or the 
extra land required for yards, stations, etc. A strip of land 
1 mile long and 8.25 feet wide contains precisely 1 acre. An 
average width of 4 rods (66 feet), therefore, requires 8 acres per 
mile. On the Boston & Albany Railroad the expenditure 
assigned to "land and land damages" a^-erages over $25000 
per mile. Of course this includes some especially expensive 
land for terminals and stations in large cities. Less than $300 
per mile was assigned to this item by an unimportant 18-mile 
road. 

355. Item 4. CLEARING AND GRUBBING. The cost of this 
may vary from zero to 100% for miles at a time, but as an 
average figure it may be taken as about 3 acres per mile at a 
cost of say S50 per acre. The possibility of obtaining valuable 
timber, wnich ma}'- be utilized for trestles, ties, or otherwise, 
and the value of which may not only repay the cost of clearing 
and grubbing, but also some of the cost of the land, should not 
be forgotten. 

356. Item 5. Earthwork. This item also includes rock- 
work. The methods of estimating the cost of earthw^ork and 
rock^\ ork. have been discussed in Chapter III. The percentage 
of this item to the total cost is very variable. On a western 
prairie it might not be more than 5 to 10%. On a road through 
the mountains it will run up to 20 or 25%,, and even more. 
The item also includes tunneling, which on some roads is a 
heavy item. 

357. Item 6. BRIDGES, TRESTLES, AND CULVERTS. This item 
will usually amount to 5 or 6% of the total cost of the road. 
In special cases, where extensive trestling is necessary, or 
several large bridges are required, the percentage will be much 
higher. On the other hand, a road whose route avoids the 
watercourses may have very little except minor culverts. On 
the Boston S: Albany the cost is given as $5860 per mile; on 
the Adirondack Railroad, S2845 per mile. Considering their 
relative character (double and single track), these figures are 
relatively what we might expect. 



396 



RAILROAD CONSTRUCTION. 



§ 358. 



358. Item 7. TRACKWORK. This item y/ill be considered as 
including everything above subgrade, except as othenvifc.e 
itemized. 

(a) Ballast. With an average width, for single track, of 
10 feet and an average of 15 inches, 2444 cubic yards of ballast 
Avill be required. The Pennsylvania Railroad estimate is 
2500 yards of gravel per mile of single track. At an estimate 
of 60 c. per yard, this costs $1500 per mile. Broken-stone 
ballast must be filled out over the ends of the ties and there- 
fore more is required; 2800 cubic yards of broken stone at 
$1.25 per yard in place will cost $3500 per mile. 

(b) Ties. Ties cost anywhere from 80 c. down to 35 c. and 
even 25 c. At an average figure of 50 c, 2640 ties per mile 
will cost $1320 per mile of single track. The cheaper ties are 
usually smaller and m^ore must be used per mile, and this tends 
to compensate the difference in cost. 

The following tabular form is convenient for reference: 

TABLE XV, NUMBER OF CROSS TIES PER MILE. 



Spacing 

center to 

center. 


Number per 
30' rail. 


Number per 
mile. 


18 inches 


20 


3520 


20 " 


18 


3168 


21 


17^ 


3017 


22.5 " 


16 


2816 


24 " 


15 


2640 


25.71" 


14 


2464 


27 " 


13^- 


2347 


30 " 


12 


2112 



(c) Rails. The total weight of the rails used per mile may 
best be seen by the tabular form. 

A convenient and useful rule to remember is that the number 
of long tons (2240 lbs.) per mile of single track equals the weight 
of the rail per yard times V". The rule is exact. For example, 
there are 3520 yards of rail in a mile of single track; at 70 lbs. 
per yard this equals 246400 lbs., or 110 long tons (exactly); 
but 70XV- = 110. 

Any calculation of the required weight of rail for a given 
weight of rolling-stock necessarily depends on the assumptions 
which are made regarding the support which the rails receive 
from the ties. This depends not only on the width and spacing 



358. 



COST OF RAILROADS. 



397 



TABLE XVI. TONS PER IMILE (WITH COST) OF RAILS OF 

VARIOUS ^VEIGHTS. 





Tons 








Tons 






Weight 


(22401b.) 


Co.st at 


Cost at 


Weight 


(22401b.) 


Cost at 


Cost at 


in lbs. 


per mile 


$26 per 


$30 per 


in lbs. 


per mile 


$26 per 


$30 per 


per yd. 


of single 
track. 


ton. 


ton. 


per yd. 


of single 
track. 


ton. 


ton. 


8 


12.571 


S326.86 


S377.14 


65 


102.143 


$2655.71 


$3064 . 29 


10 


15.714 


408 . 57 


471.43 


66 


103.714 


2696.57 


3111.43 


12 


18.857 


490.29 


565.71 


67 


105 . 286 


2737.43 


3158.59 


14 


22.000 


572.00 


660.00 


68 


106.857 


2778.29 


3205.79 


16 


25.143 


653.71 


754 . 20 


70 


110.000 


2860.00 


3300.00 


20 


31.429 


817.14 


942.86 


71 


111.571 


2900.86 


3347.14 


25 


39.286 


1021.43 


1178.57 


72 


113.143 


2941.71 


3394.29 


30 


47.143 


1225.71 


1414.29 


73 


114.714 


2982.57 


3441.43 


35 


55.000 


1430.00 


1650.00 


75 


117.857 


3064.29 


3535.71 


40 


62.857 


1634.29 


1885.71 


78 


122.571 


3186.86 


3677.14 


45 


70.714 


1838.57 


2121.43 


80 


125.714 


3268.57 


3771.43 


48 


75.429 


1961.14 


2262.86 


82 


128.857 


3350.29 


3865.71 


50 


78.571 


2042 . 86 


2357.14 


85 


133.571 


3472.86 


4007.14 


52 


81.714 


2124.57 


2451.43 


88 


138.286 


3595.43 


4148.57 


56 


88.000 


2288.00 


2640.00 


90 


141.429 


3677.14 


4242.86 


57 


89.571 


2328 . 86 


2687.14 


92 


144.571 


3758.86 


4337.14 


60 


94.286 


2451.43 


2828.57 


95 


149.286 


3881.43 


4478 . 57 


61 


95.857 


2492.29 


2875.71 


98 


154.000 


4004 . 00 


4620.00 


63 


99 . 000 


2574.00 


2970.00 


100 


157.143 


4085.71 


4714.29 



About two per cent. {2%) extra should be allowed for w^aste in cutting. 

of the ties (which are determinable), but also on the support 
which the ties receive from the ballast, which is not only very 
uncertain but variable. No general rule can therefore claim 
any degree of precision, but the following is given by the Bald- 
win Locomotive Works: " Each ten pounds weight per yard of 
ordinary steel rail, properly supported by cross-ties (not less 
than 14 per 30-foot rail), is capable of sustaining a safe load 
per wheel of 2240 pounds." For example, a consolidation loco- 
motive with 112600 lbs. on 8 dri^'ers has a load of 14075 lbs. 
per wheel. This divided b}^ 2240 gives 6.28. According to the 
rule, the rails for such a locomotive should ^^eigh at least 62.8 
lbs. per yard. 

(d) Splice-bars, track-bolts, and spikes. These are usually 
sold by the pound, except the patented forms of rail-joints, 
which are sold by the pair. In any case they are subject to 
market fluctuations in price. As an approxim.ate value the 
following prices are quoted: Splice-bars, 1.35 c. per pound; 
track-bolts, 2.4 c; spikes, 1.75 c. The v, eight of the splice- 
bars will depend on the precise pattern adopted— its cross- 
section and length. For a 45-lb. rail an angle-bar whose 
original weight in the rolled section is 6.3 lbs. per foot might 



398 



RAILROAD CONSTRUCTION. 



§ 358, 



be used. A pair 21 inches long would weigh 21.5 lbs. For a 
70-lb. rail an angle-bar section weighing 9 to 12 lbs. per yard 
would be used. A pair of the 10-lb. section, w^ith the long 
44-inch 6-hole bar, used by the Michigan Central Railroad, 
would weigh about 70 lbs. Angle-bars suitable for a 100-lb. 
rail will weigh about 14 to 16 lbs. per foot. The following 
tables will be useful for reference. 



TABLE XYII. SPT.ICE-BARS AND BOLTS PER MILE OF TRACK. 







Number of bolts 




Number 


required. 


Length 


of pairs 




of rail. 


of splice- 








bars. 


4-liole 


6-hole 






splice. 


splice. 


24 feet 


440 


1760 


2640 


25 " 


422 


1688 


2532 


26 '' 


406 


1624 


2436 


27 " 


391 


1564 


2346 


28 " 


377 


1508 


2262 


30 " 


352 


1408 


2112 


33 " 


320 


1880 


1920 



TABLE XVIII. ^RAILROAD SI'IKES. 



Size meas- 
ured under 
head. 


Average 

number 

per keg of 

200 pounds 


Ties 24" between cen- 
ters, 4 spikes per tie, 
number per Mile. 


Suitable 

weight of 

rail. 


Pounds. 


Kegs. 


5"X,V' 
5"X h" 


375 

400 
450 


5632 

5280 
4692 


28.16 
26.40 
23.46 


45 to 100 
40 " 56 
40 



TABLE XIX. TRACK-BOLTS. 
Average number in a keg of 200 pounds. 



Size of 


Square 


Hexagonal 


Suitable 


bolt. 


nut. 


nut. 


rail. 


3''xr 


250 


270 




Z\"X\" 


243 


201 




3y'xr 


236 


253 




3rxr 


229 


244 




4 "^\" 


222 


236 


50 pounds 


3y'xF 


170 


180 


and up- 


?>\" X \" 


165 


175 


ward. 


4 "X\" 


161 


170 




4V' X F' 


157 


165 




4Y'x¥' 


153 


160 


i 

i 



'^^F 



COST OF RAILROADS. 399 



(e) Track-laying. Much depends on the force of men em- 
ployed and the use of systematic methods; $528 per mile is 
the estimate employed by the Pennsylvania Railroad. $500 per 
mile is the estimate given in § 362. 

359. Item 8. BUILDINGS AND Miscellaneous structures. 
Except for rough and preliminary estimates, these items must 
be individually estimated according to the circumstances. The 
subitems include depots, engine-houses, repair-shops, v.ater- 
stations, section- and tool-houses, besides a large variety of 
smaller buildings. The structures include turn-tables, cattle- 
guards, fencing, road-crossings, overhead bridges, etc. The 
detailed estimate, given in § 362, illustrates the cost of these 
sm.aller items. 

360. Item 9. INTEREST ON CONSTRUCTION. The amount 
of capital that must be spent on a railroad before it has begun 
to earn anything is so veiy large that the interest on the cost 
during the period of construction is a very considerable item. The 
amount that must be charged to this head depends on the cur- 
rent rate of mone}^ on the time required for construction and 
on the abilit}^ of the capitalists to retain their capital where 
it will be earning something until it is actually needed to pay 
the company's obligations. Of course, it is not necessary to 
have the entire capital needed for construction on hand when 
construction com.mences. Assuming money to be worth 6%, 
that the work of construction will require one year, that the 
monej^-mxay be retained where it will earn something for an 
average period of six months after construction commences, 
or, in other words, it will be out of circulation six months before 
the road is opened for traffic and begins to earn its way, then 
vre may charge 3% on the total cost of construction. 

361. Item 10. TELEGRAPH LINES. This evidently depends 
on the scale of the road and the magnitude of the business to 
be operated. In the following estimate it is given as $200 
per mile, which evidently is intended to apply to the business 
of a small road. 

362. Detailed estimate of the cost of a line of road. The fol- 
lowing estimate was given in the Engineering News of Dec. 27, 
1900, of the cost of the Duluth, St. CJoud, Glencoe ik ]\Tankato 
Railroad, 157.2 miles long. 

The estimate is exactly as copied from the Engineering N^ews, 
There are some numerical discrepancies. Item 26 should evi- 



^ , Wooden-box culverts. 508300 ft. B.M. @S30 per M. . S15249 
5.-J 

Pile trestling. 4600 lin. ft. @ 35 c. per lin. ft 1610 

Timber trestling. 509300 ft. B.M. @ S30 per M 15279 16889 



400 RAILROAD CONSTRUCTION. § 3G2. 

deiitly be ba.sed on the sum of the first 25 items, and item 27 

on the sum of the first 26, The figures in parentheses ( ) are 
deduced from the figures given. 

1. Right-of-way; 1905.3 acres (12.12 acres per mile) @ $100 per 

acre S190530 

2. Clearing and grubbing. 144 acres (0.916 acre per mile) @ $50 

per acre 7200 

3. Earth excavation. 1907590 cu. yds. (12135 cu. yds. per mile) 

@ 15 c 286138 

4. Rock excavation. 5100 cu. yds. (32.44 cu. yds. per mile) @ 80 c. 4080 

I Iron-pipe culverts . 879840 lbs. @ 3c. per lb 26395 41644 

\ Bridge masonry: 5520 cu. yds. @, $8 per cu. yd 44160 

1 Bridges, iron, 100 spans, 2000000 lbs. @ 4 c. per lb. . . 80000 124160 

8. Cattle-guards 8750 

9. Ties (2640 per mile). 419813 (159.02 M.) @ 35 c 146935 

10. Rails (70 lbs. per yd.); 110 tons per mile, 17492.2 tons (159.02 

M.) @$26 384797 

11. Rail sidings (per yd.); llOtons per mile, 3300 tons (30 M.) @ $26 85800 

12. Switch timbers and ties 3300 

13. Spikes: 5920 lbs. per mile. 1107040 (187 M.) @ 1.75. c. per lb. 19373 

14. Splice-bars. 2635776 lbs. @ 1.35 c. per lb 35583 

15. Track-bolts (2 to joint (?)): 188458.3 lbs. @ 2.4 c. per lb 4520 

16. Track-laying 187.2 miles (^ $500 per mile 93600 

17. Ballasting- 2152 cu. yds. per mile, 402854 (187.2 M.) @ 60 c. . 241712 

18. Turn-out and switch furnishings 6450 

19. Road-crossings, 68040 ft. B.M. @ $30 per M 2041 

20. Section and tool-houses, 16 @ $800 12800 

21. Water-stations 15000 

22. Turn-tables, 6 @ $800 4800 

23. Depots, grounds, and repair-shops 78000 

24. Terminal grounds and special land damages 150000 

25. Fencing, 314 miles ($150 per mile) 47100 

26. Engineering and office expenses (5% of $1984458) 99222 

27. Interest on construction (3% of $2083680) 62510 

28. Rolling-stock ($5000 per mile) 786000 

29. Telegraph line: 157 miles @ $200 per mile 314 00 

$3060340 
Average cost per mile ready for operation, $19467. 
Approximate cost of 130 miles from St. Cloud to Duluth, estimated at 

$23000 per mile. 
Approximate cost of entire line from Albert Lea to Duluth, 287.2 miles, 

$6050340 ($21060 per mile). 



PART II. 



KAILROAD ECOIn^OMICS. 



CHAPTER XVIII. 



INTRODUCTION. 



363. The magnitude of railroad business. The gross earnings 
of railroads for the year ending June 30, 1899, were over $1,300,- 
000,000. This is greater than the combined value of all the 
gold, silver, iron, wheat, and corn produced by the country. 
The following figures (to the nearest million of dollars) gives 
the value of various crops for 1899, according to the current 
U. S. Yearbook of Agriculture: 



Gold 71 

Silver 33 

Iron 245 

Wheat 320 

Corn 629 

Total 



Oats 198 

Hay 412 

Coal. 256 

Copper 104 

Lead 19 



2287 



About 929000 persons (about one eightieth of the population) 
were directly employed by the roads for a compensation of 
about $523,000,000. Probably 3,000,000 to 4,000,000 people 
were supported by this. Beside all these, probably 5,000,000 
employes were kept bus}^ in occupations which are a more or 
less direct result of railroads, e.g., locomotive- and car-shops, 
rail-mills, etc. We may therefore estimate that perhaps 
20,000,000 people (or, say, one fourth of our population) are 
supported by railroads or b}" occupations which owe their 
chief existence to railroads. 

The ^^number of passengers carried 1 mile" was 14,591,327,613. 
Calling the population of the United States 75,000,000 for round 

401 



402 RAILROAD CONSTRUCTION. § 364. 

numbers, it means an average ride of 195 miles for every man, 
woman, and child. 

The 'Hons carried 1 mile" were 123,667,257,153, or nearly 
1650 ton-miles per inhabitant. The payments made to the 
railroads averaged over $17 per inhabitant. 

Turning to a dark side of the picture, we find that the traffic 
was carried on at a cost of 7123 killed and 44620 injured. This 
averages one killed every hour and a quarter and one injured 
every twelve minutes. Of these large numbers, the ^'passen- 
gers" comprised but 239 and 3442 respectively. The remainder 
were employes and '^ others/' the ''others" consisting largely 
of "trespassers." 

The actual bona-fide cost of the railroads of the country 
cannot be accurately computed (as will be shown later), but 
the capital, as represented by stocks and bonds, represents 
$11,033,954,898, or about $147 per inhabitant. This is roughly 
about one sixth of the total national wealth. 

The above figures may give some idea of the magnitude of 
the interests involved in the operation of railroads. No single 
business in the countr}^ approaches it in capital involved, earn- 
ings, number of people affected, or effect on other business. 

364. Cost of transportation. The importance of railroads 
may be also indicated by their power of creating cheap trans- 
portation. Less than one hundred years ago local famine 
and overabundant harvests within a radius of a few miles 
were not unknown. When the transportation of goods depended 
on actual porterage by human beings, as has been the case 
but recently in the Klondike, the transportation of 100 lbs. 20 
miles might be considered an average day's labor. At $1 per 
day, this equals $1 per ton-mile. In 1899 the railroads trans- 
ported freight at an average cost to the public of 0.724 c. per 
ton per mile, and the feeding of Europe with wheat from Mani- 
toba has become a commercial possibility. In 1899 passengers 
paid an average charge of 1.925 c. per mile, and a trip of 1000 
miles inside of 24 hours is now common. 

365. Study of railroad economics — its nature and limitations. 
The multiplicity of the elements involved in most problems 
in railroad construction preclude the possibility of a solution 
which is demonstrably perfect. Barring out the compara- 
tively few cases in this country where it is difficult to obtain 
any practicable location, it may be said that a comparatively 



§ 366. INTRODUCTION. 403 

low order of talent will suffice to locate anywhere a railroad 
over which it is physically possible to run trains. It may be 
very badly located for obtaining business, the ruling grades 
ma}^ be excessive, the alignment ma}^ be very bad, and the 
road may be a hopeless financial failure, and yet trains can be 
run. Among the infinite number of possible locations of the 
road, the engineer must determine the route which will give 
the best railroad property for the least expenditure of money — 
the road whose earning capacity is so great that after paying 
the operating expenses and interests on the bonds, the surplus 
available for dividends or improvements is a maximum. 

An unfortunate part of the problem is that even the blunders 
are not always readily apparent nor their magnitude. A de- 
fective dam or bridge will give way and every one realizes the 
failure, but a badly located railroad affects chiefly the finances 
of the enterprise b}^ a series of leaks which are only perceptible 
and demonstrable by an expert, and even he can only say that 
certain changes would probably have a certain financial value. 

366. Outline of the engineer's duties. The engineer must 
realize at the outset the nature and value of the conflicting 
interests which are involved in variable amount in each possi- 
ble route. 

(a) The maximum of business must be obtained, and yet it may 
happen that some of the business may only be obtained by an 
extravagant expenditure in building the line or by building a 
line very expensive to operate. 

(b) The ruling grades should be kept low, and yet this may 
require a sacrifice in business obtained and also viay cost more 
than it is worth. 

(c) The alignment should be made as favorable as possible; 
favorable alignment reduces the future operating expenses, 
but it may require a very large inmiediate outlay. 

(d) The total cost must be kept within the amount at which 
the earnings mil make it a profitable investment. 

(e) The road must be completed and operated until the 
'^normal" traffic is obtained and the road is self-supporting 
without exhausting the capital obtainable by the projectors : 
for no matter how valuable the property may ultimately be- 
come, the projectors will lose nearly, if not quite, all they have 
invested if the}" lose control of the enterprise before it becomes 
a paying inA^estment. 



404 RAILROAD CONSTRUCTION. § 367. 

Each new route suggested makes a new combination of the 
above conflicting elements. The engineer must select a route 
by first eliminating all hues Avhich are manifestly impracticable 
and then gradually narrowing the choice to the best routes 
whose advantages are so nearly equal that a closer detailed 
comparison is necessary. 

The ruling grade and the details of alignment have a large 
influence on the operating expenses. A large part of this 
course of instruction therefore consists of a study of operating 
expenses under average normal conditions, and then a study 
of the effect on operating expenses of given changes in the align- 
ment. 

367. Justification of such methods of computation. It may 
be argued that the data on which these computations are based 
are so unreliable (because variable and to some extent non- 
computable) that no dependence can be based on the conclu- 
sions. This is true to the extent that it is useless to, claim 
great precision in the computation of the value of any pro- 
posed change of alignment. Suppose, for example, it is com- 
puted that a given improvement in alignment will reduce the 
operating expenses of 20 trains per day by $1000 per year. 
Suppose the change in alignment may be made for $5000, which 
may be obtained at 5% interest. Even v/ith large allowances 
for inaccuracy in the computation of the value, $1000, it evi- 
dently will be better to incur an additional interest charge of 
$250 than increase the annual operating expenses by $1000. 
Moreover, since traffic is almost sure to increase (and interest 
charges are generally decreasing), the advantage of the im- 
provement will only increase as time passes. On the other 
hand, if the improvement cannot be made except by an expen- 
diture of, say, $50000, the change would evidently be unjus- 
tifiable. When the interest on the first cost is practically 
equal to the annual operating value of the proposed improve- 
ment, there is evidently but little choice; no great harm can 
result from either decision, and the decision frequently will 
depend on the willingness to increase the total amount invested 
in the enterprise. 

To express the above question more generally, in every com- 
putation of the operating value of a proposed improvement, 
it may always be shown that the true value lies somewhere 
between some maximum and some minimum. Closer calcula- 



§ 367. INTRODUCTION. 405 

tions and more reliable data will narrow the range between 
these extreme values. According as the interest on the cost 
of the proposed improvement is greater or less than the mean 
of these limits, we may judge of its advisability. The range 
of the limits shows the uncertaint}^ If it lies outside of the 
limits there is no uncertainty, assuming that the limits have 
been properly determined. If well within the limits, either 
decision will answer unless other considerations determine the 
question. And so, although it is not often possible to obtain 
precise values, we may generally reach a conclusion which is 
unquestionable. Even under the most unfavorable circum- 
stances, the computations, when made with the assistance of 
all the broad common sense and experience that can be brought 
to bear, v, ill point to a decision which is much better than mere 
'^ judgment,'' which is responsible for very many glaring and 
costly railroad blunders. In short, Railroad Economics means 
the application of systematic methods of work plus experience 
and judgment, rather than a dependence on judgment unsysv 
tematically formed. It makes no pretense to furnishing mechan- 
ical rules by which all railroad problems may be solved by any 
one, but it does give a general method of applying principles 
by which an engineer of experience and judgment can apply 
his knowledge to better advantage. To the engineer of limited 
experience the methods are invaluable; without such methods 
of work his opinions are practically worthless; with them 
his conclusions are frequently more sound than the uns3^stem- 
atically formed judgments of a man with a glittering record. 
But the engineer of great experience may use these methods 
to form the best opinions which are obtainable, for he can apply 
his experience to make any necessary local modifications in the 
method of solution. The dangers lie in the extremes, either 
recklessly applying a rule on the basis of insufficient data to 
an unwarrantable extent, or, disgusted with such evident 
unreliability, neglecting altogether such systematic methods of 
work. 



CHAPTER XIX. 

THE PROMOTION OF RAILROAD PROJECTS. 

368. Method of formation of railroad corporations. Many 
business enterprises, especially the smaller ones, are financed 
entirely by the use of money which is put into them directly 
in the form of stock or mere partnership interest. A railroad 
enterprise is frequentl}^ floated with a comparatively small 
financial expenditure on the part of the original promoters. 
The promoters become convinced that a railroad between A 
and B, passing through the intermediate towns of C and D, 
with others of less importance, will be a paying investment. 
They organize a compan^^, have surveys made, obtain a charter, 
and then, being still better able (on account of the additional 
information obtained) to exploit the financial advantages of 
their scheme, they issue a prospectus and invite subscriptions 
to bonds. Sometimes a portion of these bonds are guaranteed, 
principal and interest,, or perhaps the principal alone, by town- 
ships or by the national government. The cost of this pre- 
liminary work, although large in gross amount if the road is 
extensive, is 3^et but an insignificant proportion of the total 
amount involved. The proportionate amount that can be 
raised by means of bonds varies with the circumstances. In 
the early history of railroad building, when a road was pro- 
jected into a new country where the traffic possibilities were 
great and there was absolutely no competition, the financial 
success of the enterprise would seem so assured that no diffi- 
culty would be experienced in raising from the sale of bonds 
all the money necessary to construct and equip the road. But 
the promoters (or stockholders) must furnish all money for the 
preliminary expenses, and must make up all deficiencies be- 
tween the proceeds of the sale of the bonds and the capital needed 
for construction. 

^^In theory, stocks represent the property of the responsible 
owners of the road, and bonds are an encumbrance on that 

406 



§ 369. PROMOTION OF RAILROAD PROJECTS. 407 

propert}'. According to this theory, a railroad enterprise 
should begin with an issue of stock somewhere near the value 
of the property to be created and no more bonds should be 
issued than are absolutely necessary to complete the enter- 
prise. Xow it is not denied that there are instances in which 
this theor}^ is followed out. In New England, for example, 
as well as in some of the Southern States, there are a few roads 
represented wholly by stock or very lightly mortgaged. But 
this theory does not conform to the general history of railway 
construction in the United States, nor is it supported by the 
figures that appear in the summary. The truth is, railroads 
are built on borrowed capital, and the amount of stock that is 
issued represents in the majority of cases the difference between 
the actual cost of the undertaking and the confidence of the 
public expressed by the amount of bonds it is willing to absorb 
in the ultimate success of the venture.^' * 

^'The same general law obtains and has always obtained 
throughout the world, that such properties (as railways) are 
always built on borroAved money up to the limit of what is 
regarded as the positive and certain minimum value. The 
risk only — the dubious margin which is dependent upon sagac- 
ity, skill, and good management — is assumed and held by the 
company proper who control and manage the property." f 

369. The two classes of financial interests — the security and 
profits of each. From the above it may be seen that stocks, 
bonds, car-trust obligations, and even current liabilities repre- 
sent railroad capital. The issue of the bonds '^was one means 
of collecting the capital necessary to create the property against 
which the mortgage lies." The variation between these inter- 
ests lies chiefly in the security and profits of each. The current 
liabilities are either discharged or, as frequently happens, they 
accumulate until they are funded and thus become a definite 
part of the railroad capital. 

The growth of this tendency is shown in the following tabular 
form : 

The bonded interest has greater security than the stock, but 
less profit. The interest on the bonds must be paid before any 
money can be disbursed as dividends. If the bond interest 



* Henry C. Adams, Statistician, U. S. Int. Con. Commission. 
t A. M. Wellington, Economic Theory of Railway Location. 



408 



RAILROAD CONSTRUCTION. 



369. 



Railroads in the United 
States. 


June 30, 1888. 


June 30, 1898. 


Amount, 
millions. 


Per cent. 


Amount, 
millions. 


Per cent. 


Stocks. . 


3864 

38G9 

396 


47.5 

47.6 

4.9 


5311 
5510 
1087 


44 6 


Funded deL\- . . 


46 3 


Current liabilities, etc 


9.1 



is not paid, a receivership, and perhaps a foreclosure and sale 
of the road, is a probabilit}^, and in such case the stockholder's 
interests are frequently wiped out altogether. The bond- 
holder's real profit is frequently very different from his nomi- 
nal profit. He sometimes buys the bonds at a very considerable 
discount, which modifies the rate which the interest received 
bears to the amount really invested. Even the bondholder's 
security may suffer if his mortgage is a second (or fifth) mort- 
gage, and the foreclosure sale fails to net sufficient to satisfy 
all previous claims. 

On the other hand, the stockholder, who may have paid in 
but a small proportion of his subscription, may, if the venture 
is successful, receive a dividend which equals 50 or 100% of the 
money actually paid in, or, as before stated, his entire holdings 
may be entirely wiped out by a foreclosure sale. When the 
road is a great success and the dividends very large, additional 
issues of stock are generally made, which are distributed to the 
stockholders in proportion to their holdings, either gratuitously 
or at rates which give the stockholders a large advantage over 
outsiders. This is the process known as ^^ watering." While 
it may sometimes be considered as a legitimate ^^ salting down" 
of profits, it is frequently a cover for dishonest manipulation of 
the money market. 

For the twelve years between 1887 and 1899 about two thirds 
of all the railroad stock in the United States paid no dividends, 
while of those that paid dividends the average rate varied 
from 4.96 to 5.74%. The year from June 30, 1898, to June 30, 
1899, was the most prosperous year of the group, and yet nearly 
60% of all railroad stock paid no dividend, and the average 
rate paid by those which paid at all was 4.96%. The total 
amount distributed in dividends was greater than ever before, 
but the average rate is the least of the above group because many 
roads, which had passed their dividends for many previous 



§ 370. PROMOTION OF RAILROAD PROJECTS. 409 

years, distinguished themselves by declaring a dividend, even 
though small. During that same period but 13.35% of the 
stock paid over 6% interest. The total dividends paid amounted 
to but 2.01% of all the capital stock, while investments ordi- 
narily are expected to yield from 4 to 6% (or more) according 
to the risk. Of course the effect of '^watering" stock is to 
decrease the nominal rate of dividends, but there is no dodging 
the fact that, watered or not, even in that year of ''good times,'' 
about 60% of all the stock paid no dividends. Unfortunately 
there are no accurate statistics showing how much of the stock 
of railroads represents actual paid-in capital and how much 
is ''water." The great complication of railroad finances and 
the dishonest manipulation to which the finances of some rail- 
roads have been subjected would render such a computation 
practically worthless and hopelessly unreliable now. 

During the year ending June 30, 1898 (which may in general 
be considered as a sample), 15.82% of the funded debt paid no 
interest. About one third of the funded debt paid between 
4 and 5% interest, which is about the average which is paid. 

The income from railroads (both interest on bonds and divi- 
dends on stock) may be shown graphically by diagrams, such 
as are given in the annual reports of the Interstate Commerce 
Commission. They show that while railroad investments are 
occasionally very profitable, the average return is less than 
that of ordinary investments to the investors. The indirect 
value of railroads in building up a section of country is almost 
incalculable and is worth many times the cost of the roads. 
It is a discouraging fact that ver}^ fcvf railroads (old enough to 
have a history) have escaped the experience of a receivership, 
with the usual financial loss to the then stockholders. But 
there is probably not a railroad in existence which, however 
much a financial failure in itself, has not profited the community 
more than its cost. 

370. The small margin between profit and loss to projectors. 
When a railroad is built entirely from the funds furnished by 
its promoters (or from the sale of stock) it will generally be a 
paying investment, although the rate of payment may be very 
small. The percentage of receipts that is demanded for actual 
operating expenses is usually about 67%. The remainder will 
usually pay a reasonable interest on the total capital involved. 
But the operating expenses are frequently 90 and even 100% of 



410 RAILROAD CONSTRUCTION. § 371. 

the gross receipts. In such cases even the bondholders do not 
get their due and the stockholders have absolutely nothing. 
Therefore the stockholder's interest is very speculative. A 
comparatively small change in the business done (as is illus- 
trated numerically in § 372) will not only wipe out altogether the 
dividend — taken from the last small percentage of the total 
receipts and which may equal 50% or more of the capital stock 
actually paid in — but it may even endanger the bondholders' 
security and cause them to foreclose their mortgage. In such 
a case the stockholders' interest is usually entirely lost. It 
does not alter the essential character of the above-stated rela- 
tions that the stockholders sometimes protect themselves 
somewhat by buying bonds. By so doing they simply decrease 
their risk and also decrease the possible profit that might result 
from the investment of a given total amount of capital. 

371. Extent to which a railroad is a monopoly. It is a popu- 
lar fallacy that a railroad, when not subject to the direct com- 
petition of another road, has an absolute monopoly — that it 
controls ^'all the traffic there is" and that its income will be 
practically independent of the facilities afforded to the public. 
The growth of railroad traffic, like the use of the so-called 
necessities or luxuries of life, depends entirely on the supply 
and the cost (in money or effort) to obtain it. A large part of 
railroad traffic belongs to the unnecessary class — such as travel- 
ing for pleasure. Such traffic is very largely affected by mere 
matters of convenience, such as well-built stations, convenient 
terminals, smooth track, etc. The freight traffic is very largely 
dependent on the possibility of delivering manufactured articles 
or produce at the markets so that the total cost of production 
and transportation shall not exceed the total cost in that 
same market of similar articles obtained elsewhere. The crea- 
tion of facilities so that a factory or mine may successfully 
compete with other factories or mines will develop such traffic. 
The receipts from such a traffic may render it possible to still 
further develop facilities which will in return encourage further 
business. On the other hand, even the partial withdrawal of 
such facilities may render it impossible for the factory or mine 
to compete successfully with rivals; the traffic furnished by 
them is completely cut off and the railroad (and indirectly the 
whole community) suffers correspondingly. The ''strictly 
necessary" traffic is thus so small that few railroads could pay 



§372 



PROMOTION OF RAILROAD PROJECTS. 



411 



their operating expenses from it. The dividends of a road 
come from the last comparatively small percentage of its revenue, 
and such revenue comes from the ^'unnecessary'^ traffic which 
must be coaxed and which is so easily affected by apparently 
insignificant '^ conveniences/' 

372. Profit resulting from an increase in business done; loss 
resulting from a decrease. In a subsequent chapter it will 
be shown that a large portion of the operating expenses are 
independent of small fluctuations in the business done and that 
the operating expenses are rouglily two thirds of the gross 
revenue. Assume that by changes in the alignment the business 
obtained has been increased (or diminished) 10%. Assume for 
simplicit}^ that the operating expenses on the revised track 
are the same as on the route originally planned; also that the 
cost of the track is the same and hence the fixed charges are 
assumed to be constant for all the cases considered. Assume 
the fixed charges to be 28%. The additional bu.siness, when 
carried in cars otherwise but partly filled will hardly increase 
the operating expenses by a measurable amount. When 
extra cars or extra trains are required, the cost will increase 
up to about 60% of the average cost per train mile. We may 
say that 10% increase may in general be carried at a rate of 
40% of the average cost of the traffic. A reduction of 10% 
in traffic may be assumed to reduce expenses a similar amount. 
The effect of the chansre in business will therefore be as follows: 





Business increased 10%. 


Business decreased 10%. 


Operating exp. = G7 
Fixed charges =2S 


67(1 + 10% X 40%)= 69.68 
28.00 


67(1 - 10% X 40%)= 64.32 
28.00 








95 
Total income. . . 100 


97.68 
Income 110.00 


92.32 
Income 90 . 00 


Available for divi- 
dends 5 


Available for divi- 
dends 12.32 


Deficit 2 . 32 



In the one case the increase in business, which may often 
be obtained by judicious changes in the alignment or even by 
better management without changing the alignment, more than 
doubles the amount available for dividends. In the other case 
the profits are gone, and there is an absolute deficit. The 
above is a numerical illustration of the argument, previously 



412 RAILROAD CONSTRUCTION. § 373. 

stated, of the small margin between profit and loss to the original 
projectors. 

373. Estimation of probable volume of traffic and of probable 
growth. Sinee traffic and traffic facilities are mutually inter- 
dependent and since a large part of the normal traffic is merely 
potential until the road is built, it follows that the traffic of a 
road will not attain its normal volume until a considerable 
time after it is opened for operation. But the estimation even 
of this normal volume is a very uncertain problem. The esti- 
mate may be approached in three ways: 

1st. The actual gross revenue derived by all the railroads 
in that section of the count r}^ (as determined by State or U. S. 
Gov. reports) may be divided by the total population of the 
section and thus the average annual expenditure per head of 
population ma}^ be determined. A determination of this value 
for each one of a series of years will give an idea of the normal 
rate of growth of the traffic. Multiplying this annual contri- 
bution by the population which may be considered as tributary 
gives a valuation of the possible traffic. Such an estimate is 
unreliable (a) because the average annual contribution may not 
fit that particular locality, (h) because it is very difficult to 
correotly estimate the num^ber of the true tributary population 
especially when other railroads encroach more or less into the 
territory. Since a rough value of this sort may be readily 
determined, it has its value as a check, if for nothing else. 

2d. The actual revenue obtained by some road whose 
circumstances are as nearly as possible identical with the road 
to be considered may be computed. The weak point consists 
in the assumption that the character of the two roads is identical 
or in incorrectly estimating the allowance to be made for ob- 
served differences. The method of course has its value as a 
check. 

3d. A laborious calculation may be made from an actual 
study of the route — determining the possible output of all 
factories, mines, etc., the amount of farm produce and of lumber 
that might be shipped, with an estimate of probable passenger 
traffic based on that of like towns similarly situated. This 
method is the best when it is properly done, but there is always 
the danger of leaving out sources of income — both existent 
and that to be developed by traffic facilities, or, on the other 
hand, of overestimating the value of expected traffic. In the 



§373. 



PROMOTION OF RAILROAD PROJECTS. 



413 



following taljiilar form are shown the population, gross re- 
ceipts, receipts per head of population, mileage, earnings per 
mile of line operated, and mileage per 10,000 of population for 
the whole United States. It should be noted that the values 
are only averages, that individual variations are large, and that 
onl}^ a very rough dependence may be placed on them as applied 
to any particular case. 



Year. 


Population 
(estimated). 


Gross 
receipts. 


Receipts 
per head 
of popu- 
lation. 


Mileage, t 


Earnings 

per mile 

of line 

operated. 


Mileage 

per 
10,000 

popula- 
tion.! 


1888.. . 
1889... 
1890... 


60,100,000 

61,450,000 

*62,801,571 


$910,621,220 

964,816,129 

1051,877,632 


S15.15 
15.81 
16.75 


136,884 
153,385 
156,404 


$6853 
6290 
6725 


24.94 

25.67 
26.05 


1891.. . 
1892.. . 
1893.. . 


64,150,000 
65,500,000 
68,850,000 


1096,761,395 
1171,407,343 
1220,751,874 


17.10 
17.89 
18.26 


161,275 
162,397 
169,780 


6801 
7213 
7190 


26.28 
26.19 
26.40 


1894.. . 
1895.. . 
1896.. . 


68.200,000 
69,550,000 
70,900,000 


1073,361,797 
1075,371,462 
1150,169,376 


15.74 
15.46 
16.22 


175,691 
177,746 
181,983 


6109 
6050 
6320 


26.20 
25.97 
25.78 


1897.. . 
1898.. . 
1899.. . 
1900.. . 


72,350,000 

73.600,000 

74,950,000 

*76,295,220 


1122,089,773 
1247,325,621 
1313,610,118 
1480,673,054 


15.53 

16.95 
17.53 
19.41 


183,284 
184,648 
187,535 
190,406 


6122 
6755 
7005 
7776 


25.53 
25.32 
25.25 
24.96 



* Actual. 

t Excludes a small percentage not reporting "gross receipts." 

t Actual mileage. 

The probable growth in traffic, after the traffic has once 
attained its normal volume, is a small but almost certainquantity. 
In the above tabular form this is indicated by the gradual 
gro^i:h in ^'receipts per head of population" from 1888 to 
1893. Then the sudden drop due to the panic of 1893 is clearly 
indicated, and also the gradual growth in the last few years. 
Even in England, where the population has been nearly station- 
ary for many years, the growth though small is unmistakable. 
On the other hand the growth in some of the Western States 
has been very large. For example, the gross earnings per head 
of popula^tion in the State of Iowa increased from $1.42 in 1862 
to SIO.OO in 1870, and to $19.46 in 1884. 

There will seldom be any justification in building to accommo- 
date a larger business than what is ^'in sight." Even if it 
could be anticipated with certainty that a large increase in 



414 RAILROAD CONSTRUCTION. § 374. 

business would come in ten years, there are many reasons why 
it would be unwise to build on a scale larger than that required 
for the business to be immediately handled. Even though it 
may cost more in the future to provide the added accommo- 
dations (e.g. larger terminals, engine-houses, etc.), the extra 
expense will be nearly if not quite offset by the interest saved 
by avoiding the larger outlay for a period of years which may 
often prove much longer than was expected. A still more im- 
portant reason is the avoidance of uselessly sinking money at 
a time when every cent may be needed to insure the success 
of the enterprise as a whole. 

374. Probable number of trains per day. Increase with 
growth of traffic. The number of passenger trains per day 
cannot be determined by dividing the total number of passengers 
estimated to be carried per day by the capacity of the cars 
that can be hauled by one engine. There are many small 
railroads, running three or four passenger trains per day each 
way, w^hich do not carry as many passengers all told as are 
carried on one heavy train of a trunk line. But because the 
bulk of the passenger traffic, especially on such light-traffic 
roads, is " unnecessary'' traffic (see § 371) and must be encouraged 
and coa^xed, the trains must be run much more frequently 
than mere capacity requires. The minimum number of passen- 
ger trains per day on even the lightest-traffic road should be 
two. These need not necessarily be passenger trains exclusively. 
They may be mixed trains. 

The number required for freight service may be kept more 
nearly according to the actual tonnage to be moved. At least 
one local freight will be required, and this is apt to be considerably 
within the capacity of the engine. Some very light-traffic 
roads have little else than local freight to handle, and on such 
there is less chance of economical management. Roads with 
heavy traffic can load up each engine quite accurately according 
to its hauling capacity and the resulting economy is great. Fluc- 
tuations in traffic are readily allowed for by adding on or drop- 
ping off one or more trains. Passenger trains must be run on 
regular schedule, full or empty. Freight trains are run by 
train-despatcher's orders. A few freight trains per day ma)^ be 
run on a nominal schedule, but all others will be run as extras. 
The criterion for an increase in the number of passenger trains 
is impossible to define by set rules. Since it should always 



§ 375. PROMOTION OF RAILROAD PROJECTS. 415 

come before it is absolutely demanded by the train capacity 
being overt axed^ it may be said in general terms that a train 
should be added when it is believed that the consequent in- 
crease in facilities will cause an increase in trafhc the value of 
which will equal or exceed the added expense of the extra train. 

375. Effect on trafiic of an increase in facilities. The term 
facilities here includes everything <which facilitates the transport 
of articles from the door of the producer to the door of the 
consumer. As pointed out before, in many cases of freight 
transport, the reduction of facilities below a certain point will 
mean the entire loss of such traffic owing to local inability to 
successfully compete with more favored localities. Sometimes 
owing to a lack of facilities a railroad company feels compelled 
to pa}^ the cartage or to make a corresponding reduction on 
what would normally be the freigiit rate. In competitive freight 
business such a method of procedure is a virtual necessity in 
order to retain even a respectable share of the business. Even 
though the railroad has no direct competitor, it must if possible 
enable its customers to meet their competitors on even terms. 
In passenger business the effect of facilities is perhaps even 
more marked. The pleasure travel will be largely cut down 
if not destroyed. It is on record that a railroad company 
once ordered the manager of a station restaurant to largely 
increase the attractions at that restaurant (as a method of 
attracting traffic) and agreed to pay the expected resulting 
loss. The net result was not only a large increase in railroad 
business (as was expected), but even an increase in the profits 
of the restaurant. 

376. Loss caused by inconvenient terminals and by stations 
far removed from business centers. This is but a special case 
of the subject discussed just in the preceding paragraph. The 
competition once existing between the West Shore and the 
New York Central was hopeless for the West Shore from the 
start. The possession of a terminal at the Grand Central 
Station gave the New York Central an advantage over the West 
Shore with its inconvenient terminal at Weehawken which 
could not be compensated by any obtainable advantage by 
the West Shore. This is especially true of the passenger busi- 
ness. The through freight business passing through or termi- 
nating at New York is handled so generally by means of floats 
that the disadvantage in this respect is not so great. ThQ 



416 liAILKOAD COxMSTRUCTION. § 376. 

enormous expenditure (roughly $10,000,000) made by the 
Pennsylvania R. R., on the Broad Street Station (and its ap- 
proaches) in Philadelphia, a large part of which was made in 
crossing the Schuylkill River and running to City Hall Square, 
rather than retain their terminal in West Philadelphia, is an 
illustration of the policy of a great road on such a question. 
The fact that the original plan and expenditure has been very 
largely increased since the first construction proves that the 
management has not only approved the original large outlay, 
but saw the wisdom of making a very large increase in the ex- 
penditure. 

The construction of great terminals is comparatively infrequent 
and seldom concerns the majority of engineers. But an engineer 
has frequently to consider the question of the location of a 
way station with reference to the business center of the toT\Ti. 
The following points may (or may not) have to be considered, 
and the real question consists in striking a proper balance 
between conflicting considerations. 

(1) During the early history of a railroad enterprise it is 
especially needful to avoid or at least postpone all expenditures 
which are not demonstrably justifiable. 

(2) The ideal place for a railroad station is a location im- 
mediately contiguous to the business center of the toA\Ti. The 
location of the station even one fourth of a mile from this may 
result in a loss of business. Increase this distance to one mile 
and the loss is very serious. Increase it to five miles and the 
loss approaches 100%. 

(3) The cost of the ideal location and the necessary right 
of way may be a very large sum of money for the new enterprise. 
On the other hand the increase in property values and in the 
general prosperity of the town, caused by the railroad itself, 
will so enhance the value of a more convenient location that its 
cost at some future time will generally be extravagant if not 
absolutely prohibitory. The original location is therefore under 
ordinary conditions a finality. 

(4) To some extent the railroad will cause a movement of 
the business center toward it, especially in the establishment 
of new business, factories, etc., but the disadvantages caused 
to business already established is permanent. 

(5) In any attempt to compute the loss resulting from a 
location at a given distance from the business center it must be 



§ 377. PROMOTION OF RAILROAD PROJECTS. 417 

recognized that each problem is distinct in itself and that any 
change or growth in the business of the town changes the amount 
of this loss. 

The argument for locating the station at some distance from 
the center of the town may be based on (a) the cost of right 
of way, thus involving the question of a large initial outla}^, 
(h) the cost of very expensive construction (e.g. bridges), 
again invoh^ng a large initial outlay, (c) the avoidance of ex- 
cessive grade into and out of the town. It sometimes happens 
that a railroad is follomng a line which would naturalh^ cause 
it to pass at a considerable elevation above (rarely below) 
the town. In this case there is to be considered not only the 
possible greater initial cost, but the even more important increase 
in operating cost due to the introduction of a very heavy grade. 
To study such a case, compute the annual increase in operating 
expenses due to the additional grade, curvature, and distance; 
add to this the annual interest on the increased initial cost 
(if any) and compare this sum with the estimated annual loss 
due to the inconvenient location. The estimation of the increase 
in operating expenses is discussed in a subsequent chapter. 
The loss of business due to inconvenient location can only be 
guessed at. Wellington says that at a distance of one mile 
the loss would average 25%, wdth upper and lower limits of 
10 and 40%, depending on the keenness of the competition 
and other modifying circumstances. For each additional mile 
reduce 25% of the preceding value. While such estimates are 
grossly approximate, yet with the aid of sound judgment they 
are better than nothing and may be used to check gross errors. 

377. General principles which should govern the expenditure 
of money for railroad purposes. It will be shown later that 
the elimination of grade, curvature, and distance have a positive 
money value ; that the reduction of ruling grade is of far greater 
value ; that the creation of facilities for the handling of a large 
trafhc is of the highest importance and yet the added cost of 
these improvements is sometimes a large percentage of the 
cost of some road over which it w^ould be physically possible 
to run trains between the termini. 

The subsequent chapters will be largely devoted to a discussion 
of the value of these details, but the general principles governing 
the expenditure of money for such purposes may be stated as 
follows; 






418 RAILROAD CONSTRUCTION. § 377. 

1. No money should be spent (beyond the unavoidable 
minimum) unless it may be shoA\Ti that the addition is in itself 
a profitable investment. The additional sum may not Avreck 
the enterprise and it may add something to the value of the 
road, but unless it adds more than the improvement costs it is 
not justifiable. 

2. If it may be positively demonstrated that an improvement 
will be more valuable to the road than its cost, it should certainly 
be made even if the required capital is obtained with difficulty. 
This is all the more necessary if the neglect to do so will per- 
manently hamper the road with an operating disadvantage 
which will only grow worse as the traffic increases. 

3. This last principle has two exceptions: (a) the cost of 
the improvement may wreck the whole enterprise and cause 
a total loss to the original investors. For, unless the original 
promoters can build the road and operate it until its stock 
has a market value and the road is beyond immediate danger 
of a receivership, they are apt to lose the most if not all of 
their investment; (h) an improvement which is very costly 
although unquestionably wise may often be postponed by means 
of a cheap temporary construction. Cases in point are found 
at many of the changes of alignment of the Pennsylvania R. R., 
the N. Y., N. H. & H. R. R., and many others. While some of 
the cases indicate faulty original construction, at many of the 
places the original construction was wise, considering the then 
scanty traffic, and now the improvement is wise considering 
the great traffic. 



CHAPTER XX. 

OPERATING EXPENSES. 

378. Distribution of gross revenue. When a railroad com- 
prises but one single property, owned and operated bv itself, 
the distribution of the gross revenue is a comparatively simple 
matter. The operating expenses then absorb about two thirds 
of the gross revenue; the fixed charges (chiefly the interest on 
the bonds) require about 25 or 30% more, leaA^ng perhaps 3 
to 8% (more or less) available for dividends. A recent report 
on the Fitchburg R. R. shows the following: 

Operating expenses $5,083,571 69 . 1% 

Fixed charges 1,567,640 21.3% 

Available for dividends, surplus, or per- 
manent improvements 708,259 9.6% 

Total revenue $7,359,470 100 . 0% 

But the financial statements of a large majority of the railroad 
corporations are by no means so simple. The great consolida- 
tions and reorganizations of recent years have been effected 
by an exceedingly complicated system of leases and sub-leases, 
purchases, '' mergers, '^ etc., whose forms are various. Railroads 
in their corporate capacity frequently o^^tl stocks and bonds 
of other corporations (railroad properties and otherwise) and 
receive, as part of their income, the dividends (or bond interest) 
from the investments. 

In consequence of this complication, the U. S. Interstate 
Commerce Commission presents a ''condensed income account'' 
of which the following is a sample (1899): 

Gross earning-3 from operation (received by 

station-agents, etc) $1,313,610,118 

Less operating expenses (fuel, wages, etc.) 856.968,999 

Income from operation 456,641,119 

Income from other sources (lease of road, stocks, 

bonds, etc.) 148,713,983 

Total income 605,355,102 

419 



420 RAILROAD CONSTRUCTION. § 378. 

Total deductions from income (interest, rents for 

lease of road, taxes, etc.) 441 200,289 

Net income ' 164,154!813 

Total dividends (including "other payments")-. 111,089,936 

Surplus from operations 53,064,87.7 

In the above account an item of income (e.g. lease of road) 
reported by one road will be reported as a ^'deduction from 
income" by the road which leases the other. 

The above statement may be reduced to an income account 
of all the raihvays considered as one system. We then have 

Operating expenses $856,968,999 

Salaries and maintenance of leased lines 
(really operating expenses, but con- 
sidered above as fixed charges against 
the leasing Unes) 595,192 

857,564,191 64.1% 

Net interest and taxes 295,098,014 22 .0% 

Available for dividends, adjustments, 

and improvements 186,992,909 13.9%, 

1,339,655,114 100.0%) 

Gross earnings from operation 1,313,610,118 

Clear income from investments (i.e., 
the balance of intercorporate pay- 
ments and receipts on corporate in- 
vestments) 26,044,996 

1.339,655,114 

Of the $186,992,909, the amount disbursed as dividends to 
outside stockholders (besides that paid to railroads in their 
corporate capacity) was S94,273,796. This left a balance of 
$92,719,113 ^'available for adjustments and improvements.'' 
Of this part was spent in permanent improvements, part was 
advanced to cover deficits in the operation of weak lines and 
more than half was left as ^'surplus/' i.e, working capital. 

The percentages of the gross revenue which are devoted to 
operating expenses, fixed charges, and dividends are not neces- 
sarily an indication of creditable management or the reverse. 
Causes utterly beyond the control of the management, such 
as the local price of coal, may abnormally increase certain 
items of expense, while ruinous competition may cut down the 
gross revenue so that little or nothing is left for dividends. 
A favorable location will sometimes make a road prosperous 



§ 379. OPERATING EXPENSES. 421 

in spite of bad management. On the other hand, the highest 
grade of skill will fail to keep some roads out of the hands of 
a receiver. 

379. Fourfold distribution of operating expensc^^ The 

distribution of operating expenses here used is copied from the 
method of the Interstate Commerce Commission. The aim is to 
divide the expenses into groups which are as mutually indepen- 
dent and distinct as possible — although, as will be seen later, 
a change in one item of expense will variously affect other 
items. The groups are : 

Average value, 

1. Maintenance of way and structures 20.662^ 

The values for five years have an extreme range of 
about 1.2%, The subdivisions of this group and of 
the others will be given later. 

2. Maintenance of equipment 16.892^ 

Extreme range of 1.834%. The tendency has been 
for this item to grow larger, not only in absolute amount 
but in percentage of total expenditure. 

3. Conducting transportation 57.793^ 

This item has been growing relatively less. During 
(and immediately after) the panic of 1893, the main- 
tenance of way and of equipment was made as small 
as possible, which made the cost of conducting trans- 
portation relatively larger. During the recent more 
prosperous years deficiencies of equipment have been 
made up, making this item relatively less. 

4. General expenses 4 . 653^ 

A nearly constant item. 

100.000^ 

The above percentages represent the averages given by the 
reports for the five years from 1895 to 1899 inclusive. 

380. Operating expenses per train-mile. The reports of the 
U. S. Interstate Commerce Commission give the average cost 
per train-mile for every railroad in the United States. Although 
there are wide variations in these values, it is remarkable that 
the very large majority of roads give values which agree to 
within a small range, and that within this range are found not 
onh' the great trunk lines with their enormous train mileage, 
but also roads -^^^ith xevy light traffic. 

In the follo^ang tabular form is shown a statement taken from 
the report for 1898 of ten of the longest railroads in the United 
States and, in comparison with them, a corresponding statement 



422 



RAILROAD CONSTRUCTION. 



§ 380. 



regarding ten more roads selected at random, except in the 
respect^ that each had a mileage of less than 100 miles. Al- 
though the extreme variations are greater, yet there is no very 
marked difference in the general values for operating expenses 
per train-mile, or in the ratio of expenses to earnings. The 
averages for the ten long roads agree fairly well with the averages 
for the whole country, but there would be no trouble (as is 
shown by some of the individual cases) in finding another group 
of ten short roads giving either greater or less average values than 
those given. And yet the tendency to uniform values, regard- 
less of the mileage, is very striking. 



No. in 
report. 




Mileage. 


Operat- 
ing ex- 
penses 
per train- 
mile. 


Ratio ^''^: 
earn. 




Whole United States 


186,396 


0.956 


65 58 








71 


Canadian Pacific 


6.568 
6,191 
5,860 
5,426 
5,232 
5,086 
4,565 
4,524 
3,860 
3,807 


0.854 
0.883 
0.881 
1.320 
0.809 
0.885 
0.917 
1.177 
1.101 
.764 


58 31 


1465 


C.,M. & St. P 


58 94 


1443 


C, B., &Q 


60 87 


1879 


Southern Pacific 


58.70 


1142 
1436 
1405 


Southern 

Chicago & Northwestern 

A.,T. & S. F 


65.32 
63.35 
67.59 


1560 


Northern Pacific. . . . 


46 81 


1495 


Great Northern. . . . . 


46 97 


1264 


Ilhnois Central 

Average of ten . . 


63.56 






0.969 


59 . 04 











7 
105 
167 
234 

888 
1074 
1284 
1540 
1812 
1979 



Bennington & Rutland. 

Mont. & Wells R 

Balto. & Del. Bay 

Cent. N. Y. & W 

Man. & N. E 

Farmv. & Powh 

Lex. & East 

Manistique 

Wh. & Bl. River Val. . 
No. Pac. Coast 



Average of ten 



59 


0.582 


44 


0.828 


45 


1.098 


63 


0.454 


99 


0.739 


93 


0.781 


94 


0.975 


60 


1.162 


64 


.799 


88 


.769 




0.819 





71.42 
83.96 
102.83 
91.17 
54.49 
76.22 
68.46 
69.01 
53.08 
66.58 



73.72 



The constancy of the average cost per train mile for several 
years past may be noted from the following tabular form. 

The enforced economies after the panic of 1893 are well 
showTi. The reduction generally took the form of a lowering 
of the standards of maintenance of way and of maintenance of 



§381, 



OPERATIXG EXPENSES. 



423 



Year. 


Average cost per 
train-mile. 


1890 

1891 


96.006 
95.707 
96.580 
97.272 
93.478 
91.829 
93 . 838 
92.918 
95.635 
98 . 390 


■ 1892. 


1893 

1894 

1895. 


1896 

1897 

1898 


1899 




95.165 



equipment. The marked advance from 1897 to 1898 and to 
1899 was largeh^ caused by the necessity for restoring the roads 
to proper condition, replenishing worn-out equipment and pro- 
viding additional equipment to handle the greatly increased 
volume of business. 

In looking over the list, it may be noted that the cases where 
the operating expenses per train-mile and the ratio of expenses 
to earnings vary ver^^ greatly from the average are almost 
invariably those of the very small roads or of ^'junction roads" 
where the operating conditions are abnormal. For example, 
one little road, with a total length of 13 miles and total annual 
operating expenses of $5342, spent but 22^c. per train-mile, 
which precisely exhausted its earnings. As another abnormal 
case, a road 44 miles long spent $3.81 per train-mile, which was 
nearly fourteen times its earnings. In another case a road 
13 miles long earned S7.76 per train-mile and spent $6.03 (78%) 
on operating expenses, but the fixed charges were abnormal 
and the earnings were less than half the sum of the operating 
expenses and fixed charges. The normal case, even for the 
small road, is that the cost per train-mile and the ratio of operat- 
ing expenses to earnings will agree fairly well with the average, 
and when there is a marked difference it is generally due to 
some abnormal conditions of expenses or of earning capacity. 

381. Reasons for uniformity in expenses per train-mile. 
The chief reason is that, although on the heavy -traffic road 
everything is kept up on a finer scale, better roadbed, heavier 
rails, better rolling stock, more employees, better buildings, 
stations, and terminals, etc., yet the mmiber of trains is so nmch 
greater that the divisor is just enough larger to make the average 



424 liAILROAb CONSTRtrCtlON. § 382. 

cost about constant. This is but a general statement of a fact 
which will be discussed in detail under the different items of 
expense. 

382. Detailed classification of expenses with ratios to the 
total expense. The Interstate Commerce Commission now 
publishes each year a classification with detailed summation 
for the cost of each item. These summations are made up 
from reports furnished by railroads which have (in the reports 
already made) represented about 94% of the total traffic han- 
dled. In the annexed tabular form (Table XX) are shown the 
percentages which each item bears to the total. The character of 
the changes from year to year in these ratios is very instructive 
and will be considered in the detailed discussion of the items 
which will follow. 

Table XX is copied from the Interstate Commerce Commis- 
sion report for 1899, pp. 88-90. 

383. Elements of the cost (with variations and tendencies) 
of the various items. The I. C. C. report for the year ending 
June 30, 1895, was the first to include the distribution of ex- 
penses according to the present classification. The number of 
reports since then are too few to be of much value in determining 
the tendency to variation of the several items, and similar 
calculations made in previous years have by no means an equal 
reliability. Nevertheless the items as given are reliable and 
may be utilized, as far as any such computations are to be 
depended on, in estimating future expenses. A great deal of 
very interesting and instructive information may be derived 
from a study of the variations of these items, but the chief 
purpose of this discussion is to point out those elements of 
the cost of operating trains which may be affected by such 
changes of location as an engineer is able to make. There are 
some items of expense with which the engineer has not the 
slightest concern — nor will they be altered by any change in 
alignment or constructive detail which he may make. In the 
following discussion such items will be passed over with a brief 
discussion of the sub-items included. 



MAINTENANCE OF WAY. 

384. Item I. Repairs of Roadway. The item of repairs 
of roadway is very large — about half of the total cost of main- 



TABLE XX. SUMMAR 



nAILROADS IN THE UNITED STATES FOR THE TEAR ENDING 
DR THE YEARS ENDING JUNE 30, 1899 TO 1895. 



UNK 30, ISOn, AND 



Item. 


Amount. 




Per cen 






1899. 


1899. 


1898.* 


1897.t 


1896.} 


1895.5 


Maintenance of way and structure: 


Ills 

19.335.860 
3.968,408 
17,762,120 

liissS 

3,628.539 


1;i 

2.374 
.487 
2.181 

11 

:44e 


Hi 

2.512 
.637 
1.967 

:| 

:349 


10.644 

lit! 

2.472 
.609 
1.745 

11 

:3i8 

20.972 


10.738 
litt 
2.205 
.661 
1.794 

11 
:372 


10 236 


llS&-v--------------- 


UT. 


4. Kepairs an,l renewals of bridges an< 

■ 5. IkpLus"'^,.;! n.iK.wals'oi'fenceV.'roa.' 

7. l:.-|.air., i,i„rr.ricu'als"oi;"docks"and 

8. Ilcijairs anil'renewalaor telegraph.' '.'//. 


2.268 
.520 
1.648 

11 
1304 


Total 


8169,825,0.54 


19.824 


Maintenance of equipment 

11 Superintendence 

12 1 (| nil, in 1 uiipii 1 Is of locomotives 

. 1 ; ! \ 'r?STc^ir- 

I 1 1 morkcars 

II 1 1 .f marine equip 

ill 1 k of shop machinery 
19 Other expeMM """""* 


S0.47586 

lii 

■"l 708 416 

4 167 798 
4 429 987 


7 038 
210 


li 

159 

242 

486 
038 
19! 


li- 

b 376 

213 


'IS 

173 
(140 


b'.B60 
.4.55 


Total 


SI43 294 445 


17 ol, 


17 3,9|16 .02 




1.5.701 



I S766 332 900 excluding '^51 640 376 unclassified 

«|,92 191 617 excluding 860 033 127 unclassified 

S721 730 766 excluding S51 268 278 unclassified 

467)228 640 excluding ?60 491775 unilassificd 



Telegraph expenses.. 

Station service 

Station supplies 

bXnce! 
if equipment, bi 



Loss and e 

Injuries to persons 

Clearing wrecks 

Operating marine equipment 

Advertising 

Outside agencies 

Commissions 

Stockyards and elevators. . . . 
Rents for tracks, yards, and 

Rents for buildings and < 



§14,392,891 
78,913,978 
77,187.344 
5,038,615 



12,439,075 

33.791, .383 
15. .525 .232 
01.100.732 



3,.509,073 
14,507.499 
1,580,909 



3,967,353 
5,107,066 
5,456,377 



I39S.» 1897.t| 1896.t 1895.5 



7!600] 
1.525 



( General expenses. 

47. Salaries of general officers 

48. Salaries of clerks and attendants. 

49. General office expenses and sup- 


89,535,486 
10,864,401 

3:b32:sSS 

1,390,670 
3,838.087 

.536,819,710 


1:33^ 
.292 
!7lo 

All 
~752i 


i.'sli 

.285 
1656 

4.419 


.301 
.165 


1.213 
1.409 

:44( 
:34'4 


im 




isOG 
.172 




52. S'tatioiiery and printing (general 




Total 


4.613 


4.955 




Hopapitutetion of expenses 

55. Mainl?n;u,rVof eqiiiiiinpMtV.'. ; 

57: tif'nCTTuNpelSwl"'''"'.'",'';; ■ 


'il'1'1 


.57 n:il 
4, 521 


II 




20.634 
4.613 


59.460 
4.9.55 


Gr.ind tntaU 


•SM.3S9.709 


UIO.OUO 




100.000 


100.000 


100.000 



I Excludes $42,579,197 



424 SAlLROAb CONStRtJCtlON. ^-i^ 

cost about constant. This is but i gener 1 stit nent of a f 
which will be discussed in detail under tl e d fterent ten 
expense. 

382. Detailed classification of expenses with ratios to t 
total expense. The Interstate Commerce Comnuss on n 
publishes each year a classification w th deta led summ t 
for the cost of each item. These summat ons 1 e mad 
from reports furnished by railroads h cl 1 e ( tl p 
already made) represented about 94^ of tl t 1 t II | 
died. In the anne.ved tabular for (Tall \\) I 
percentages which each item bears to tl c t t I II 1 | 
the changes from year to year in tl ese rat o n t 
and will be considered in the deta led d scu s on of tl e I 
which will follow. 

Table XX is copied from the Ii te state Con r e Co 
sioii report for 1899, pp. 88-90. 

383. Elements of the cost (with var at ons and tenden 
of the various items. The I. C. C rej t tor tl e >ear e I 
June 30, 189.5, was the first to incl d tl e d st I ut of 
penses according to the present cl fi t o i 1 e n b 
reports since then are too few to be of uch al e n d t n 
the tendency to variation of the sev ral to n I 
calculations made in previous years ha e I y no I 
reliability. Nevertheless the items as g ve Ml 
may be utilized, as far as any such co [ utat o t 
depended on, in estimating future expe es \ g t tl I 
very interesting and instructive nfornato n aj be d 
from a study of the variations of tl e e terns L t (1 
purpose of this discussion is to po nt t tl ose ele e t 
the cost of operating trains whicl mv 1 e atfe ted by 
changes of location as an engineer able to ake The 
some items of expense with whicl tl e eng e h s not I 
.slightest concern — nor wiU they be Itered 1 j a cl n 
alignment or constructive detail \ 1 cl 1 e } uake I t 
following discussion such items will be pas.cd o\ er n tl a b 
discussion of the sub-items included. 



384. Item I. Repairs of Roadway. The item of repairs 
of roadway is very large— about half of the total cost of miO' 



§ 385. OPERATING EXPENSES 425 

tenance of way and structures. It includes the cost of frogs, 
switches, switch-stands, and interlocking signals. The dis- 
tribution and laying of ties and rails, ballasting and tamping 
track, ditching, weeding, widening and protecting banks, the 
maintenance of snow-fences, dikes, and retaining walls, are also 
included. In short, an}^ expense of maintaining the roadbed 
in condition which cannot be definitely assigned to one of the 
next few items will generally belong to this item — except per- 
haps those of item 10 (g.v.). The larger part of such items of 
expense is labor, and the variations will largely depend on the 
fluctuations in the wages of trackmen. Formerly these were 
much higher than now. About fifteen years ago they had 
dropped to what Wellington considered to be a permanent 
average of $1.25 per day. In 1893 it had dropped to sSl.22, 
then in 1897 and 1898 to s$1.16. In 1899 it was raised to $1.18. 

In 1899 the average cost of this item per mile of main track 
was about $480, but this figure, after all, is of but little value 
because, for the reason already given in general in §381, it will be 
found that the cost for any road varies almost exactly as the 
train-mileage and will average very closely to lie. per train- 
mile, whether the traffic be heavy or light. 

385. Item 2. Renewal of Rails ; This item may be con- 
sidered as having been withdrawn from the previous item 
simply because it is one of the largest of the single items and 
because its cost is very readily determined. It includes the 
cost of the rails, their inspection, and their delivery (but not 
their distribution). The item shows a large percentage of vari- 
ation, the figures (percentage of total expenses) being 1.322, 
1.391, 1.546, 1.444, and 1.499 by the last five reports. The 
drop from 1.546 in 1897 to 1.391 in 1898 was just 10%. These 
fluctuations are due first to that considerable fluctuation in the 
price of rails which railroads can hardly expect to escape, and 
secondly to variations in the standard of maintenance caused 
first by hard times, which are then followed by unusual expen- 
ditures in good times, or by the expenditures absolutely essen- 
tial to restore the track to its former condition. The item 
includes all rails wherever used, whether on main track, siding, 
repair track, gravel track, on wharves or coal-docks, and even 
includes guard-rails. But it does not include any rail attachments 
such as joints, frogs, switches, etc. The rate of rail wear under 
various conditions has already been discussed in Chapter IX. 



426 RAILROAD CONSTRUCTION. § 386. 

386. Item 3. Renewal of Ties. As with the previous 
item, this item is simply a detachment from the general item, 
repairs of roadway. As with rails, the cost of laying and dis- 
tributing the ties is not included, but the cost of tie-plates and 
tie-plugs, also chemical treatment for preservation, if such is 
used, is included in this item. While the cost will vary con- 
siderably between different roads on account of first cost, kind 
of wood, climate, etc., the item for any one road for a period of 
years cannot vary greatly, unless there is a marked change in 
the standard of maintenance. The actual cost of such work 
has already been discussed in Chapter VIII. 

387. Item 4. Repairs and Renewals of Bridges and 
Culverts. This item includes not only the maintenance cost 
of all ]:)ridges, trestles, viaducts, and cuh'erts, but of all piers, 
abutments, riprapping, etc., necessary to maintain them, and 
even the cost of operating drawbridges. The locating engi- 
neer is not concerned with this item, except as he may con- 
sider that some distance which is to be added (or cut out) has 
the average number of culverts and bridges. With culverts 
and small bridges there would be little or no error in such an 
assumption, but if there were any large bridges on the portion 
of track under discussion, they would need special consideration. 

388. Items 5 to 10. Repairs and Renewals of Fences, 
Road Crossings, and Cattle-guards — Of Buildings and 
FIXTURES — Of Docks and Wharves — Of Telegraph Plant; 
Stationery and Printing; and "Other Expenses." These 
items in the aggregate amount to but 3% of the average cost 
per train-mile. The fluctuations have so small an effect on the 
average cost per train-mile that they may be neglected. In 
item 5 are included not only those things which are specifically 
mentioned, but also those structures which in general are not 
directly affected by the running of trains. For example, '^ road 
crossings'^ include not only the maintenance of highway cross- 
ings at grade, but also overhead highway crossings and what- 
ever a railroad may have to pay for the maintenance of a bridge 
by which another railroad crosses it. On the other hand, the 
maintenance of a bridge by which a railroad crosses another 
road (highway or railroad) is charged to bridges. The effect 
(if any) of these items on any changes in construction which 
an engineer may make will be specifically discussed in the suc- 
ceeding chapters. 



? 389. OPERATIXG EXPENSES. 427 



MAINTENANCE OF EQUIPMENT. 

389. Item II. Superintendence i This item includes those 
fixed charges in superintendence which do not fluctuate with 
small variations in business done. It includes the salaries 
of superintendent of motive power, master mechanic, master 
car-builder, foremen, etc., but does not include that of road 
foremen of engines nor enginemen. In a general way the item 
is proportional to the general scale of business of the road, but 
does not fluctuate with it. 

390. Item 12. Repairs and Renewals of Locomotives; 
This item must be studied b}" the locating engineer in order to 
determine the effect on locomotive repairs and renewals of an ad- 
dition to distance (considered in Chapter XXI), the effect (chiefly 
in wheel wear) of a reduction in curvature (considered in Chap- 
ter XXII), or the effect of grade (considered in Chapter XXIII). 
In studying the effect of grade, the policy of adopting heavier 
locomotives and the effect of this on this item must also be 
considered. This item includes the expenses of work whose 
effect is supposed to last for an indefinite period. It does not 
include the expense of cleaning out boilers, packing cylinders, 
etc., which occurs regularly and which is charged to item 21, 
round-house men. It does include all current repairs, general 
OA^erhauling, and even the replacement of old and worn-out 
locomotives by new ones to the extent of keeping up the original 
standard and number. Of course additions beyond this must 
be considered as so much increase in the original capital invest- 
ment. As a locomotive becomes older the annual repair charge 
becomes a larger percentage on the first cost, and it may be- 
come as much as one fourth and even one third of the first cost. 
^\Tien a locomotive is in this condition it is usually consigned to 
the scrap-pile; the annual cost for maintenance becomes too 
large an item for its annual mileage. The effect on expenses 
of increasing the weight of engines is too complicated a prob- 
lem to admit of precise solution, but certain elements of it may 
be readily computed. While the cost of repairs is greater for 
the heavier engines, the increase is only about one half as fast 
as the increase in weight — some of the sub-items not being 
increased at all. 



428 RAILROAD CONSTRUCTION. § 391. 

391. Items 13, 14, 15. Repairs and Renewals of Pas- 
senger Cars, of Freight Cars, and of Work Cars. As 
with engine repairs, the item excludes consumable supplies (oil, 
waste, illuminating oil or gas, ice, etc.), but includes in general 
all items necessary to maintain the cars up to the full standard 
of condition and number, and even to replace old worn-out 
cars by new. When, as is frequently the case with both cars 
and locomotives, the new rolling stock is larger, better, and of a 
higher standard th5.n that which is replaced, the difference in 
cost should be added to capital investment. The chief con- 
cern of the locating engineer regarding this item is the effect 
on car repairs of additional distance, of variations in curva- 
ture (affecting wheel wear chiefly), and of grade (affecting the 
draft-gear and general wear and tear). These items w^ill be 
considered under their proper heads in the folloAving chapters. 

392. Items 16, 17, 18, and 19. REPAIRS AND RENEWALS 

OF MARINE Equipment — Of Shop Machinery and Tools; 
Stationery and Printing; Other Expenses. The location 
of the road along the line has no connection with the main- 
tenance of marine equipment. The maintenance of shop 
machinery and tools can only be affected as the w^ork of repairs 
of rolling slock fluctuates, and of course in a much smaller 
ratio. No change Avhich an engineer can effect will have any 
appreciable influence on this item. 

The other items are too small and have too little connection 
with location to be here discussed except as it may be considered 
that they vary with train mileage, which an engineer may 
influence (see Chapter XXIII, Grades). 



conducting transportation. 

393. Item 20. Superintendence. As with item 11, this 
item is not subject to minor fluctuations in business, but only 
varies with changes in the general scale of the business of the 
road. 

394. Item 21. Engine and Round-house Men. This item 

includes the wages of engineers, firemen, and also all men em- 
ployed around the engine-houses except those who are making 
such repairs as should be charged to maintenance of equipment 
(item 12). The item is a large one, but is only affected by one 
class of change of location — a difference in length of line. The 



I § 395. OPERATING EXPENSES. 429 

wages of the round-house men constitute but a small percentage 

of this item, and the wages of the enginemen vary almost directly 

as the mileage. On very short roads, where the number of 
I round trips which may proper!}^ constitute a da^^'s work is 

definitely limited and on which there is but little night or Sunda}^ 
i work, the wages may be practically by the day, and a variation 
1] in length of several hundred feet or even a few miles in the 

length of the road may make practically no difference in the 
ij wages paid. But on the larger roads, operated by divisions, 
ji on which (especially in freight work) there is no distinction of 
; day or night, Aveek day or Sunday, the varying length of divisions 
•lis equalized by calling them IJ or IJ runs, a ^'run" usually 
I being considered as about 100 miles. The enginemen are then 
i paid according to the number of runs made per month. The 

effect on this item of variations in distance is discussed more 

fully in Chapter XXT. 

395. Item 22. Fuel for Locomotives. The item includes 

the entire cost of the fuel until it is placed in the engine-tender. 
jThe cost therefore includes not only the first cost at the point 
' of delivery to the road, but also the expense of hauling it over 
the road from the point of delivery to the various coaling stations 
and the cost of operating the coal -pockets from which it is 
loaded on to the tenders. Although the cost is fairly regular 
I for any one road, it is exceedingly variable for different roads. 
Roads running through the coal regions can often obtain their 
coal for eighty or ninety cents per ton. Other roads far re- 
moved from the coal-mines have been compelled to pay six dollars 
per ton. In the three succeeding chapters there will be con- 
, sidered in detail the effect on fuel consumption of variations 
(in location. It will be shoTSTi that fuel consumption is quite 
largely independent of distance and the number of cars hauled. 

396. Items 23, 24, and 25. WATER-SUPPLY ; OIL, TALLOW, 
I AND Waste ; Other Supplies for Locomotives: The cost of 

the water-supply is quite largeh^ a fixed charge except where 
it is supplied by municipalities at meter rates. The consump- 
tion of all these supplies will vary nearly as the engine-mileage. 

397. Item 26. Train Service. This item is one of the 
^ largest single items and includes in general the wages of all 
I the train-hands except the enginemen. As with enginemen, 
(they are paid according to the number of runs. The item is 

therefore of importance to the locating engineer from the one 



li 



430 RAILROAD CONSTRUCTION. § 398. 

standpoint of distance, and even then only when the variation 
in distance which is considered will affect the classification of 
the run and therefore the rate of pay for that run. 

398. Item 27. Train Supplies and Expenses. These in- 
clude the large list of consumable suppHes such as lubricating 
oil, illuminating oil or gas, ice, fuel for heating, cleaning materials, 
etc., which are used on the cars, and not on the locomotives. 
The consumption of some of these articles is chiefly a matter 
of time ; — in other cases it is a function of the mileage. .; 

399. Items 28, 29, 30, and 31. SWITCHMEN, FLAGMEN, AND 

Watchmen; Telegraph Expenses; Station Service; and 
Station Supplies. These items will be proportional to the 
general scale of business of the road, but are independent of small 
fluctuations in business. The main items are obvious from the 
titles. Many sub-items, which are ver}^ small or are of occasional 
or accidental occurrence, are also included under these items for 
'lack of a better classification. 

400. Items 32,33, and 34. SWITCHING CHARGES — Balance; 
Car Mileage — Balance; Hire of Equipment. The first of 
these is a charge paid by a road to other corporations for 
s^\"itching done for the road. The locating engineer is not 
concerned with this item. 

Car Mileage. This is a charge paid by a road for the use 
of the cars (chiefly freight cars) of another road. To save the 
rehandling of freight at junctions the policy of running freight 
cars on to foreign roads is very extensively adopted. Since 
the foreign road receives (ultimately) its mileage proportion 
of the freight charge, it justly pays the home road a rate which 
is supposed to represent the value of the use of a freight car 
for so many miles. The foreign road then loads up the freight 
car with freight consigned to some point on the home road and 
sends it back, again paying mileage for the distance traveled 
on the foreign road, a proportional freight charge having been 
received for that service. By a clearing-house arrangement 
the various roads settle their debit and credit accounts ^^dth 
each other by the payment of a balance. Such is the simple 
theory. In practice the cars are not sent back to the home 
road at once, but wander off according to the local demand. 
As long as strict account is kept of the movements of every 
car and the home road is paid a charge which really covers 
the value of such service, no harm is done the home road except 



i§401. OPERATING EXPENSES. 431 

j that sometimes, when business has suddenly increased, the 
home road cannot get enough cars to handle its business. The 
1 value of a car is then abnormally above its ordinary value 
! and the home road suffers for lack of the rolling stock which 
belongs to it. The charge being paid according to mileage, 
,i any variations of distance have a direct bearing on this item. 
f Hire of Equipment. This may refer to locomotives or cars 
; which are hired for a special service, or, on very poor roads, 
'! it may refer to equipment, Avhich is hired rather than purchased. 
\ The locating engineer has no concern with this item. 

401. Items 35, 36, and 37. LOSS AND DAMAGE; INJURIES 
TO Persons; Clearing Wrecks. These expenses are fortuitous 
and bear no absolute relation to road-mileage or train-mileage. 
While they depend largely on the standards of discipline on 

, the road, even the best of roads have to pa}^ some small pro- 
portion of their earnings to these items. The possible relation 
between curvature and accidents is discussed in Chapter XXII, 

I but otherwise the locating engineer has no concern with these 

> items. 

402. Items 38 to 53. All of the remaining items (for a list 
of which see § 382) are of no concern to the locating engineer. 
They are either general expenses (such as taxes) or are special 

, items (such as the operation of marine equipment) which will 
not be changed by variations in distance, curvature, or grades 
which a locating engineer may make. They will not therefore 
be further discussed. 



CHAPTER XXI. 
DISTANCE. 

403. Relation of distance to rates and expenses. Rates 
are usually based on distance traveled, on the apparent 
hypotheses that each additional mile of distance adds its pro- 
portional amount not only to the service rendered but also to 
the expense of rendering it. Neither hypothesis is true. The 
value of the service of transporting a passenger or a ton of 
freight from A to ^ is a more or less uncertain gross amount 
depending on the necessities of the case and independent of 
the exact distance. Except for that very small part of passen- 
ger traffic which is undertaken for the mere pleasure of traveling, 
the general object to be attained in either passenger or freight 
traffic is the transportation from A to B, however it is attained. 
A mile greater distance does not improve the service rendered ; 
in fact, it consumes valuable time of the passengers and perhaps 
deteriorates the freight. From the standpoint of service ren- 
dered, the railroad which adopts a more costly construction and 
thereby saves a mile or more in the route between two places 
is thereby fairly entitled to additional compensation rather 
than have it cut down as it would be by a strict mileage rate. 
The actual vjilue of the service rendered may therefore vary 
from an insignificant amount which is less than any reasonable 
charge (which therefore discourages such traffic) and its value 
in cases of necessity — a value which can hardly be measured in 
money. If the passenger charge between New^ York and Phila- 
delphia were raised to $5, SIO, or even $20, there Avould still be^ 
some passengers who would pay it and go, because to them 
it would be worth $5, $10, or $20, or even more. Therefore, 
when they pay $2.50 they are not pa^-ing what the service is 
worth to them. The service rendered cannot therefore be 
made a measure of the charge, nor is the service rendered pro- 
portional to the miles of distance. 

The idea that the cost of transportation is proportional to 

432 



§ 404. DISTANCE. 433 

the distance is much more prevalent and is in some respects 
more justifiable, but it is still far from true. This is especially 
true of passenger service. The cost of transporting a single 
passenger is but little more than the cost of printing his ticket. 
Once aboard the train, it makes but little difference to the 
railroad whether he travels one mile or a hundred. Of course 
there are certain very large expenses due to the passenger 
traffic which must be paid for by a tariff which is rightfully 
demanded, but such expenses have no relation to the cost of an 
additional mile or so of distance inserted between stations. 
The same is true to a slightly less degree of the freight traffic. 
As shown later, the items of expense in the total cost of a train- 
mile, which are directly affected by a small increase in distance, 
are but a small proportion of the total cost. 

404. The conditions other than distance that affect the cost; 
reasons why rates are usually based on distance. Curvature 
and minor grades have a considerable influence on the cost of 
transportation, as will be shown in detail in the next two chap- 
ters, but they are never considered in making rates. Ruling 
grades have a very large influence on the cost, but they are Hke- 
wise disregarded in making rates. An accurate measure of 
the effect of these elements is difficult and complicated and 
w^ould not be appreciated by the general public. Mere dis- 
tance is easily calculated; the public is satisfied with such 
a method of calculation; and the railroads therefore adopt a 
tariff which pays expenses and profits even though the charges 
are not in accordance with the expenses or the service rendered. 

An addition to the length of the line may (and generally does) 
involve curvature and grade as well as added distance. In 
this chapter is considered merely the effect of the added dis- 
tance. The effect of grade and curvature must be considered 
separately, according to the methods outlined in succeeding 
chapters. The additional length considered is likewise assumed 
not to affect the business done nor the number of stations, but 
that it is a mere addition to length of track. 



EFFECT OF DISTANCE ON OPERATING EXPENSES. 

405. Effect of slight changes in distance on maintenance of 
way. With a few unimportant exceptions all the items of 
i expense under maintenance of way and structures (see § 407) 



434 RAILROAD CONSTRUCTION. § 406. 

will be increased directly as any increase in distance. This 
must certainly be true for items 1, 2, 3, and 5, which alone 
comprise about three fourths of the total expense for mainte- 
nance of way. If we assume that the proposed change of length 
involves no difference in the number of bridges, culverts, build- 
ings, and fixtures, docks and wharves, we may consider items 
4, 6, and 7 to be unaffected. This will generally be true for 
small changes in length, measured in feet. For larger differ- 
ences, measured in miles, items 4 and 6 will vary nearly as the 
distance. The same may be said of items 9 and 10. The cost 
of maintaining the telegraph line will probably be increased 
about 60% of the unit cost. The effect of changes in distance 
on these various items of maintenance of way (as well as the 
other items of expense of a train-mile) will be tabulated in § 408. 

406. Effect on maintenance of equipment. The relation 
between an increase in length of line and the expenses of items 
11, 15, 17, 18, and 19 are quite indefinite. In some respects 
they would be unaffected by slight changes of distance. From 
other points of view there is no reason why the expenses should 
not be considered proportionate to the distance. For exam- 
ple, the added track will probably require as much work from 
the construction ti'ain as any other part of the road and is 
therefore responsible for as much of the ^'repairs and renewals 
of work-cars" — item 15. Fortunately all of these items are so 
small, even in the aggregate, that little error will be involved 
by either decision. It will therefore be assumed that these 
items are affected 100% for large additions in distance and but 
50% for small additions. 

Item 16 is evidently unaffected. 

Item 12. Locomotives deteriorate (1) with age; (2) by 
expansion and contraction, especially of the fire-boxes, when 
fires are drawn and relighted; (3) on account of the strains due 
to stopping and starting; (4) the strains and wear of wheels due 
to curved track; (5) the additional stresses due to grade and 
change of grade; and (6) on account of the work of pulling 
on a straight level track. Obser^^e that the first five causes 
have no direct relation to an addition of mere distance (the 
possible curvature or grade incident to the additional distance 
being a separate matter). How much of the total deteriora- 
tion is due to the last cause? Wellington attacks this problem 
as follows: the records of engine-repair shops readily furnish 



§ 407. DISTANCE. 435 

the proportionate cost of the repairs of boiler, running-gear, 
etc. An estimate is then made of the effect of each cause on 
each item. For example, the boiler is responsible for 20% of 
the repairs and renewals. Of this 7% (ssiy one third) is assigned 
to '^terminal service, getting up steam, making up trains," 
4% to curvature and grades, 2% to '' stopping and starting 
at way stations," and the other 7% to '^distance on tangent 
between stations." The other items are treated similarly. 
Wellington says, " As this [subdivision of expenses] has been 
done with great care to get the best attainable authority for 
each (which it would occupy too much space to give in detail), 
the margin for possible error is not great enough to be of mo- 
ment, although no absolute exactness can be claimed for it." 
His final estimate is that distance is responsible for 42% of 
the total cost of repairs and renewals. This value will there- 
fore be used for all additional distances, great cr small. 

Items 13 and 14. The causes of deterioration of both passenger 
and freight cars may be classified exactly as above — omitting 
merely cause 2 — the expansion and contraction due to firing. 
Considering that a large part of the repairs of freight cars is 
due to the draft-gear and brakes, which are affected chiefly 
by the heavy strains due to stopping and starting and to grades, 
while the repairs of wheels are largely due to the wear of wheels 
on curves, it is not surprising that he allows only 36% of the 
cost of repairs and renew^als of freight cars to be due to straight 
distance. He made no direct estimxate for passenger-cars, but 
points out the fact that the maintenance of the seats, furniture, 
and ornamentation make up much more than half the cost 
of passenger-car repairs. A large part of such deterioration 
is due to age and the weather, although that of the seats is 
largely a function of passenger wear and therefore of distance 
traveled. Although the items of deterioration in passenger 
cars is ver}- different from those of freight cars. 3'et if a similar 
calculation is made for passenger cars it will be found that the 
final figure is substantially the same as for freight cars and will 
here be so regarded. 

407. Effect on conducting transportation. Item 20. This is 
evidently unaffected by small or even considerable additions 
to distance. 

Item 21. Theoretically, train wages should var}^ as mileage. 
On the larger roads, where, especially in the freight service, 



436 RAILROAD CONSTRUCTION. § 407. 

there is little or no distinction of day or night, week-day or Sun- 
day, it is practically impossible to hire the trainmen to work 
between certain definite hours of the day and pa}^ them accord- 
ingly, as is done with factory emplo3^ees. As explained in Chap- 
ter XX, § 394, the system usually adopted of paying trainmen is 
such that small changes of distance (measured in feet) would 
not affect train wages. The wages of round-house men would 
not be affected under an}^ conditions, and those of the enginemen 
and of the trainmen (item 26) w^ould not generall}^ be affected 
unless the change of distance is very great — perhaps ten miles. 
Since items 21 and 26 are both very large, it will not do to 
ignore this item or to average it. The pay of round-house men 
is about 7% of item 21. We may therefore say that if the 
change in distance is so great that trains wages will be affected, 
item 21 will be affected 93% and item 26 will be affected 100%. 
For shorter changes of distance they will be unaffected. 

Item 22. A surprisingly large percentage of the fuel con- 
sumed is not utilized in drawing a train along the road. Part 
of this loss is due to firing up, part is wasted when the engine 
is standing still, which is a large part of the total time. The 
policy of banking fires instead of draAving them reduces the 
injury resulting from great fluctuations in temperature, but 
the total coal consumed is about the same and we may there- 
fore consider that almost a fireboxful of coal is wasted whether 
the fires are banked or drawn. The amount thus wasted (or at 
least not utilized in direct hauling) has been estimated at 5 to 
10% of the whole consumption. Experiments* have shown that 
an engine standing idle in a yard, protected from wind, well 
jacketed, etc., will require from 25 to 32 lbs. of coal per hour 
simply to keep up steam. It has been found that the fastest 
express trains will lose one fourth of their total time between 
termini in stops, and freight trains on a single-track road will 
generally spend four hours per day on sidings. The waste of 
coal from this cause is estimated at 3 to 6% of the total con- 
sumption. The energy consumed in stopping and starting is 
very great. A train running 30 miles per hour has enough 
kinetic energy to move it on a level straight track more than 
two miles. Ever}^ time a train running at 30 miles per hour 
is stopped, enough energy is consumed by the brakes to run 

* Wellington, Economic Theory, p. 207. 



§ 407. DISTANCE. 437 

it from one to two miles. When starting, it will require an 
equal amount of work to restore that velocit}^, in addition to 
the ordinary resistances. It has been shown that on the Man- 
hattan Elevated Railroad, w^here stops will average every three 
eighths of a mile, this cause alone wall account for the consump- 
tion of nearly three fourths of the fuel. Of course on ordinary 
railroads the proportion is not nearly so great, but it is probably 
as much as 10 to 2v0% as an average figure. For a through 
express train making but few stops the figure would be small, 
except for the effect of ^'slow-downs." For suburban trains 
the proportion* would be abnormally high. The fuel required 
to overcome the added resistances due to curvature and grade 
are of course exceedingly variable, depending on the particular 
alignment of the road considered. An approach to the truth 
may be made by considering the average curvature per mile 
for the roads of the United States and the average grades, 
and computing, by the methods given in subsequent chapters, 
the extra fuel consumed on account of such average conditions, 
and these items will apparently be responsible for 3% due to 
curvature and about 15% due to grades. Summarizing the 
above we have: 

Firing 5 to 10% 

Wasted while still Z" 6% 

Stopping and starting 10 ' ' 20% 

Average curvature 3'' 3% 

Average grade 15 ' ' 15% 

36 54 
Direct hauling 64 ' ^ 46 Average, 55% 

100 100 

This shows that the addition of mere straight level distance 
would not increase the consumption of fuel more than 55% of 
the average consumption per mile. 

Items 23, 24, and 25. If water is paid for by meter, the cost 
is strictly according to consumption, which w^ould vary almost 
according to the number of engine-miles. When supplied 
from the company's own plant, as is usually the case, a slight 
increase will not appreciably affect the cost. Nothing is wasted 
during firing or while the engine is still. The use is therefore 
more nearly as the mileage, and the cost for an additional mile 



438 RAILROAD CONSTRUCTION. § 408. 

may be considered as 50% of its average cost per train-mile. 
Items 24 and 25 will be considered similarly. Fortunately 
these items, whose variation with additional distance is some- 
what obscure and variable, only aggregate a little over 1% of 
the cost of a train-mile and therefore a considerable percentage 
of error is of little or no importance. 

Item 26. (See comments on item 21.) 

Item 27. This item, as well as many other small items that 
follow, will be irregularly affected by a small increase in distance. 
It would appear equally wrong to say that they would be un- 
affected or to say that they will vary directly as the mileage. 
50% will be allowed. 

Item 28. The necessity for flagmen and watchmen varies 
in general as the mileage. An addition in distance is less apt 
to increase the number of switchmen*. 50% of this item will 
be added. 

Item 29. Telegraph expenses include the wages of operators 
(unaffected), and the special expenses due to offices and tele- 
graph stations and to operating the line — the maintenance of 
the line being charged to item 8. This item will be but little 
affected, if at all, by additional distance, but 20% will be allowed. 
Items 30, 31, 32, and 34 are unaffected. Items 33, 35, 36, and 
37 are affected 100%. Items 38 to 48 are unaffected. 

The ^^ general expenses" (items 46 to 53) will be unaffected. 

408. Estimate of total effect on expenses of small changes 
in distance (measured in feet); estimate for distances measured 
in miles. According to the accompanying compilation the cost 
of operating additional distance will be about 35% of the 
average cost per train-mile when the additional distance is small, 
but will be about 56% if the additional distance is several miles. 
The figures may also be considered as the saving in the oper- 
ating expenses resulting from a shortening of the line. 

The average cost of a train- mile during the years from 1890 to 
1899 varied from 91.8c. to 98.4c., with an average value of 95.2c. 
On this basis the above figures become 33.2 and 53.3 cents per 
train-mile respectively. Some trains run 365 days per year, 
others but 313. The tendency is toward the larger figure and 
it will therefore be used in these calculations. The added cost 
per daily train per year for each foot of distance is 

33.2X365X2 . _ 
5280 -^•^^^- 



408. 



DISTANCE. 



439 



When the distance is measured by miles the added cost per 
daily train per year for each mile of distance is: 

53.3x36oX2=S389. 



Table xxi. — Effect on operating expenses of great 
(and small) changes in distance. 





6 


Per cent 
affected. 


Cost per mile. 




6 


Per cent 
affected. 


Cost per mile. 


• 


> 
























7^ 










;z; 


03 












^ 










F 


03 










1— 


a 

O 






Great. 


Small. 




a 




-M 


03 

P3 


Great. 


Small. 




iz; 


O 


m 








1^ 





m 




















22.454 






15.04 


5.94 


1 


10.596 


100 


100 


10.60 


10.60 


26 


7.722 


100 





7.72 





2 


1.440 


100 


100 


1.32 


1.32 


27 


1.528 


50 


50 


.76 


.76 


3 


3.093 


100 


100 


3.09 


3.09 


28 


4.136 


50 


50 


2.07 


2.07 


4 


2.378 


100 





2.38 





29 


1.974 


20 


20 


.39 


.39 


5 


.523 


100 


100 


.52 


.52 


30 


7.818 














6 


1.865 


30 





.56 


0| 


31 


.762 














7 


.247 














32 


.345 














8 


.135 


60 


60 


.08 


.08 


33 


2.094 


100 


100 


2.09 


2.09 


9 


.027 


100 





.03 





34 


.333 














10 


.358 


100 





.36 


01 


35 


.738 


100 


100 


.74 


.74 














36 
37 
38 
39 


.883 
.131 
.887 
.426 


100 
100 

1 


100 
100 


.88 
.13 


.88 
.13 




20.662 




.... 


18.94 


15.61 










11 


.650 


50 





.32 











12 


5 . 879 


42 


42 


2.47 


2.47 


40 


1.692 










13 


2.209 


36 


36 


.80 


.80 


41 


.171 


1 








14 


6.765 


36 


36 


2.44 


2.44 


42 


.130 


!- 











15 


. 155 


100 


50 


.15 


.08 


43 


1.848 










16 


.209 














44 


.492 










17 


.490 


100 


50 


.49 


.25! 


45 


.610 










18 


.040 


100 


50 


.04 


.02 


46 


.619 


J 








19 


.495 


100 


50 


.50 


.25 
















57.793 






29.82 


13.00 




16.892 






7.21 


6.31 










.... 




47 
48 


1 










20 


1.761 






















21 


9.781 


93 





9.10 





49 


1 










22 


9.681 


55 


55 


5.32 


5.32 


50 


M.653 














23 


.671 


50 


50 


.34 


.34 


51 












24 


.276 


50 


50 


.19 


.19 


52 












25 


.184 


50 


50 


.09 


.09 1 


53 


1 












22.454 






15.04 


5.94 

1 


100.000 






55.97 


34.92 















Light-traffic roads are more apt to run their trains on week 
days only, and a corresponding reduction should be made in 
these cases. 

Regarding the accuracy of the above computations, it should 
be noted that the most uncertain items are generally the smallest, 
and that even the largest variations that can reasonably be 



440 RAILROAD CONSTRUCTION. § 409. 

made of the above figures will not very greatly alter the final 
result. A numerical illustration of the value of saving distance 
vnll be given later. 

EFFECT OF DISTANCE ON RECEIPTS. 

409. Classification of traffic. There are various methods 
of classifying traffic, according to the use it is intended to 
make of the classification. The method here adopted will have 
reference to its competitive or non-competitive character and 
also to the method of division of the receipts on through traffic. 
Traffic may be classified first as ^ through ^' and ^^ocal" — 
through traffic being that traveling over two (or more) lines, 
no matter how short or non-competitive it may be; ^' local' ^ 
traffic io that confined entirely to one road. A fivefold classifica- 
tion is however necessary — which is : 

A. Non-competitive local — on one road with no choice of 
routes. 

B. Non-competitive through — on two (or more) roads, but 
with no choice. 

C. Competitive local — a choice of two (or more) routes, but 
the entire haul may be made on the home road. 

D. Competitive through — direct competition between two 
or more routes each passing over two or more lines. 

E. Semi-competitive through— a non-competitive haul on the 
home road and a competitive haul on foreign roads. 

There are other possible combinations, but they all reduce to 
one of the above forms so far as their essential effect is concerned. 

410. Method of division of through rates between the 
roads run over. Through rates are divided between the 
roads run over in proportion to the mileage. There may 
be terminal charges and possibly other more or less arbitrary 
deductions to be taken from the total amount received, but 
when the final division is made the remainder is divided accord- 
ing to the mileage. On account of this method of division and 
also because non-competitive rates are always fixed according 
to the distance, there results the unusual feature that, unlike 
curvature and grade, there is a compensating advantage in 
increased distance, which applies to all the above kinds of 
traffic except one (competitive local), and that the compensation 
is sometimes sufficient to make the added distance an actual 



§411. DISTANCE. 441 

source of profit. It has just been proved that the cost of hauling 
a train an additional mile is only 35 to 56% of the average 
cost. Therefore in all non-competitive business (local or 
through) where the rate is according to the distance, there is 
an actual profit in all such added distance. In competitive 
local business, in which the rate is fixed by competition and 
has practically no relation to distance, any additional distance 
is dead loss. In competitive through business the profit or 
loss depends on the distances involved. This may best be 
demonstrated by examples. 

411. Effect of a change in the length of the home road on 
its receipts from through competitive traffic. Suppose the 
home road is 100 miles long and the foreign road is 150 miles 

long. Then the home road mil receive --— — -— r =40% of the 

J-UU "T~ J-OU 

through rate. 

Suppose the home road is lengthened 5 miles; then it mil 

105 
receive -7- — ——=41.176% of the through rate. The traffic 
lUo + loU 

being competitive, the rate will be a fixed quantity regardless 
of this change of distance. By the first plan the rate received 
is 0.4% per mile; adding 5 miles, the rate for the original 100 
miles may be considered the same as before; and that the addi- 
tional 5 miles receive 1 . 176%, or 0. 235% per mile. This is 59% 
of the original rate per mile, and since this is more than the 
cost per mile for the additional distance (see § 408), the added 
distance is e\idently in this case a source of distinct profit. 
On the other hand, if the line is shortened 5 miles, it may be 
similarly shown that not only are the receipts lessened, but 
that the saving in operating expenses b}^ the shorter distance 
is less than the reduction in receipts. 

A second example will be considered to illustrate another 
phase. Suppose the home road is 200 miles long and the foreign 
road is 50 miles long. In this case the home road mil receive 

900 -1- g^n "^^^/^ ^^ ^^^ through rate. Suppose the home road is 

205 

lengthened 5 miles ; then it will receive ^ -- = 80 . 392% 

.^Uo -H ou 

of the through rate. B}^ the first plan the rate received is 

0.400% per mile; adding 5 miles, there is a surplus of 0.392, 

or 0.0784 per mile, which is but 19.6% of the original rate. 



442 KAILROAD CONSTRUCTION. § 412. 

At this rate the extra distance evidently is not profitable, al- 
though it is not a dead loss — there is some compensation. 

412. The most advantageous conditions for roads forming 
part of a through competitive route. From the above it may 
be seen that when a road is but a short link in a long com- 
petitive through route, an addition to its length will increase 
its receipts and increase them more than the addition to the 
operating expenses. 

As the proportionate length of the home road increases the 
less will this advantage become, until at some proportion an 
increase in distance will just pay for itself. As the proportionate 
length grows greater the advantage becomes a disadvantage 
until, when the competitive haul is entirely on the home road, 
any increase in distance becomes a net loss without any com- 
pensation. It is therefore advantageous for a road to be a 
short link in a long competitive route; an increase in that link 
will be financially advantageous; if the total length is less than 
that of the competing line, the advantage is still greater, for 
then the rate received per mile will be greater. 

413. Effect of the variations in the length of haul and the 
classes of the business actually done. The above distances 
refer to particular lengths of haul and are not necessarily the 
total lengths of the road. Each station on the road has 
traffic relations vv^ith an indefinite number of traffic points 
all over the country. The traffic between each station on 
the road and any other station in the country between which 
traffic may pass therefore furnishes a new combination, the 
effect of which will be an element in the total effect of a 
change of distance. In consequence of this, any exact solution 
of such a problem becomes impracticable, but a sufficiently 
accurate solution for all practical purposes is frequently ob- 
tainable. For it frequently happens that the great bulk of a 
road's business is non-competitive, or, on the other hand, it 
may be competitive-through, and that the proportion of one 
or more definite kinds of traffic is so large as to overshadow 
the other miscellaneous traffic. In such cases an approximate 
but sufficiently accurate solution is possible- 

414. vjenerai conclusions regarding a eUan^e in distance, 
(a) In all non-competitive business (local and through) the 
added distance is actually profitable. Sometimes practically^ 



' S 415. DISTANCE. 443 

j all of the business of the road is non-competitive ; a considerable 
I proportion of it is always non-competitive. 

' (b) When the competitive local business is very large and the 
I competitive through business has a very large average home 
i haul compared with the foreign haul, the added distance is 
a source of loss. Such situations are unusual and are generally 
I confined to trunk lines. 

(c) The above may be still further condensed to the general 
j conclusion that there is always some compensation for the added 

cost of operating an added length of line and that it frequently 
is a source of actual profit. 

(d) There is, however, a limitation which should not be lost 
sight of. The above argument may be carried to the logical 
conclusion that, if added distance is profitable, the engineer 
should purposely lengthen the line. But added distance means 
added operating expenses. A sufficient tariff to meet these is a 
tax on the community — a tax which more or less discourages 
traffic. It is contrary to public policy to burden a community 
with an avoidable expense. But, on the other hand, a railroad 
is not a charitable organization, but a money-making enter- 
prise, and cannot be expected to unduly load up its first cost 
in order that subsequent operating expenses may be unduly 
cheapened and the tariff unduly lowered. A common reason 
for increased distance is the saving of the first cost of a very 
expensive although shorter line. 

(e) Finally, although there is a considerable and uncom- 
pensated loss resulting from curvature and grade which will 
justify a considerable expenditure to avoid them, there is by 
no means as much justification to incur additional expenditure 
to avoid distance. Of course needless lengthening should be 
avoided. A moderate expenditure to shorten the line maj^ be 
justifiable, but large expenditures to decrease distance are 
never justifiable except w^ien the great bulk of the traffic is 
exceedingly heavy and is competitive. 

415. Justification of decreasing distance to save time. It 
should be recalled that the changes which an engineer may 
make which are physically or financial^ possible will ordi- 
narily have but little effect on the time required for a trip. 
The time which can thus be saved will have practically no value 
for the freight business — at least any value which would justify 
changing the route. When there is a large directly competitive 



444 RAILROAD CONSTRUCTION. § 416* 

passenger traffic between two cities {e.g. New York to Phila- 
delphia) a difference of even 10 minutes in the time required 
for a run might have considerable financial importance, but 
such cases are comparatively rare. It may therefore be con- 
cluded that the value of the time saved by shortening distance 
will not ordinarily be a justification for increased expense to 
accomplish it. 

416. Effect of change of distance on the business done. 
The above discussion is based on the assumption that the busi- 
ness done is unaffected by any proposed change in distance. 
If a proposed reduction in distance involves a loss of business 
obtained, it is almost certainly unwise. But if by increasing 
the distance the original cost of the road is decreased (because 
the construction is of less expensive character), if the receipts 
are greater, and are increased still more by an increase in busi- 
ness done, then the change is probably wise. While it is almost 
impossible in a subject of such complexity to give a general 
rule, the f olloT\dng is generally safe : Adopt a route of such length 
that the annual traffic per mile of line is a maximum. This 
statement may be improved by allowing the element of original 
cost to enter and say, adopt a route of such length that the annual 
traffic per mile of line divided by the average cost per mile is 
a maximum. Even in the above the operating cost per mile, 
as affected by the curvature and grades on the various routes, 
does not enter, but any attempt to formulate a general rule 
which would allow for variable operating expenses would evi- 
dently be too complicated for practical application. 



CHAPTER XXII. 
CURVATURE. 

417. General objections to curvature. In the popular mind 
cun^ature is one of the most objectionable features of railroad 
alignment. The cause of this is plain. The objectionable 
qualities are on the surface, and are apparent to the non-tech- 
nical mind. They may be itemized as follows: 

1. CXirvature increases operating expenses by increasing (a) 
the required tractive force, (b) the wear and tear of roadbed 
and track, (c) the wear and tear of equipment, and (d) the 
required number of track- walkers and wa,tchmen. 

2. It may affect the operation of trains (a) by limiting the 
length of trains, and (b) by preventing the use of the heaviest 
types of engines. 

3. It may affect travel (a) b}^ the difficulty of making time, 
(b) on account of rough riding, and (c) on account of the appre- 
hension of danger. 

4. There is actually an increased danger of collision, derail- 
ment, or other form of accident. 

Some of these objections are quite definite and their true 
value may be computed. Others are more general and vague 
and are usually exaggerated. These objections will be dis- 
cussed in inverse order. 

418. Financial value of the danger of accident due to curva- 
ture. At the outset it should be realized that in general the 
problem is not one of curvature vs. no curv^ature, but simply 
sharp curvature vs. easier curvature (the central angle remain- 
ing the same), or a great-er or less percentage of elimination 
of the degrees of central angle. A straight road between ter- 
mini is in general a financial (if not a ph3^sical) impossibility. 
The practical question is then, how much is the financial value 
of such diminution of danger that may result from such elimi- 
nations of curs^ature as an engineer is able to make? 

445 



446 RAILROAD CONSTRUCTION. § 419. 

In the year 1898 there were 2228 railroad accidents reported 
by the Railroad Gazette^ whose Hsts of all accidents worth re- 
porting are very complete. Of these a very large proportion 
clearly had no relation whatever to curvature. But suppose 
we assume that 50% (or 1114 accidents) were directly caused 
by curvature. Since there are approximately 200,000 curves 
on the railroads of the country, there was on the average an 
accident for every 179 curves during the year. Therefore we 
may say, according to the theory of probabilities, that the 
chances are even that an accident may happen on an}^ particular 
curve in 179 years. This assumes all curves to be equally danger- 
ous, which is not true, but we may temporarily consider it to be 
true. If, at the time of the construction of the road, SI. 00 were 
placed at compound interest at 5% for 179 years, it would pro- 
duce in that time S620.89 for each dollar saved, wherewith to pay 
all damages, while the amount necessary to eliminate that cur- 
vature, even if it were possible, would probably be several thou- 
sand dollars. The number of passengers carried one mile for 
one killed in 1898-99 was 61,051,580. If a passenger were to 
ride continuously at the rate of sixty miles per hour, day and 
night, year after year, he would need to ride for more than 116 
years before he had covered such a mileage, and even then the 
probabilities of his death being due to curvature or to such a 
reduction of curvature as an engineer might accomplish are 
very small. Of course particular curves are often, for special 
reasons, a source of danger and justify the employment of 
special watchmen. They would also justify very large expen- 
ditures for thier elimination if possible. But as a general 
proposition it is evidently impossible to assign a definite money 
value to the danger of a serious accident happening on a par- 
ticular curve which has no special elements of danger. 

Another element of safety on curved track is that trait of 
human nature to exercise greater care where the danger is more 
apparent. Many accidents are on record which have been 
caused by a carelessness of locomotive engineers od a straight 
track when the extra watchfulness usually observed on a curved 
track would have avoided them. 

419. Effect of curvature on travel, (a) Difficulty in making 
time. The growing use of transition curves has largely elimi- 
nated the necessity for reducing speed on curves, and even ^ hen 
the speed is reduced it is done so easily and quickly by means 



§ 420. CURVATURE. 447 

of air-brakes that but little time is lost. If two parallel lines 
were competing sharply for passenger traffic, the handicap ot 
sharp curvature on one road and easy curvature on the other 
might have a considerable financial value, but ordinarily the 
7nere redvction of time due to sharp curvature will not have any 
computable financial value. 

(b) On account of rough riding. Again, this is mj.ich reduced 
by the use of transition curves. Some roads suffer from a gen- 
eral reputation for crookedness, but in such cases the excessive 
curvature is practically unavoidable. This cause probably 
does have some effect in influencing competitive passenger 
traffic. 

(c) On account of the apprehension of danger. This doubtless 
has its influence in deterring travel. The amount of its influence 
is hardly computable. When the track is in good condition 
and transition curves are used so that the riding is smooth, 
even the apprehension of danger will largely disappear. 

Travel is doubtless more or less affected by curvature, but 
it is impossible to say how much. Nevertheless the engineer 
should not ordinarily give this item any financial weight what- 
soever. Freight traffic (two thirds of the total) is unaffected 
by it. It chiefly affects that limited class of sharply competi- 
tive passenger traffic — a traffic of which most roads have not a 
trace. 

420. Effect on operation of trains, (a) Limiting the length 
of trains. When curvature actually limits the length of trains, 
as is sometimes true, the objection is valid and serious. But 
this can generally be avoided. If a curve occurs on a ruling 
grade without a reduction of the grade sufficient to compensate 
for the curvature, then the resistance on that curve will be a 
maximum and that curve will limit the trains to even a less 
weight than that which may be hauled on the ruling grade. 
In such cases the unquestionably correct policy is to ''com- 
pensate for curvature, " as explained later (see §§ 427, 428), and 
not allow such an objection to exist. It is possible for curvature 
to limit the length of trains even without the effect of grade. 
On the Hudson River R. R. the total net fall from Albany to 
New York is so small that it has practically no influence in 
determining grade. On the other hand, a considerable portion 
of the route follows a steep rock}' river bank which is so crooked 
that much curvature is unavoidable and very sharp curvature 



448 K4ILR0AD CONSTRUCTION. § 420. 

can only be avoided by very large expenditure. As a consequence 
sharp curvature has been used and the resistance on the curves 
is far greater than that of any fluctuations of grade which it 
was necessar}^ tc use. Or, at least, a comparatively small 
expenditure would Sevflice to cut down any grade so that its 
resistance would be less than that of some curve w^hich could 
not be avoided except at an enormous cost. And as a result, 
since the length of trains is really limited by curvature, minor 
grades of 0.3 to 0.5% have been freely introduced which 
might be removed at comparatively small expense The above 
case is very unusual. Low grades are usually associated with 
generally level country where curvature is easily avoided — 
as in the Camden and Atlantic R. R. Even in the extreme 
case of the Hudson River road the maximum curvature is 
only equivalent to a comparatively low ruling grade. 

(b) Preventing the use of the heaviest types of engines. The 
validity of this objection depends somewhat on the degree of 
curvature and the detailed construction of the engine. While 
some types of engines might have difficulty on curves of ex- 
tremely short radius, yet the objection is ordinarily invalid. 
This will best be appreciated when it is recalled that the ^' Con- 
solidation" type was originally designed for use on the sharp 
curvature of the mountain divisions of the Lehigh Valley R. R., 
and that the type has been found so satisfactory that it has 
been extensively employed elsewhere. It should also be re- 
membered that during the Civil War an immense traffic daily 
passed over a hastily constructed trestle near Petersburg, Va., 
the track having a radius of 50 feet. As a result of a test made 
at Renovo on the Philadelphia and Erie R. R. by Mr. Isaac 
Dripps, Gen. Mast. Mech., in 1875,* it was claimed that a 
Consolidation engine encountered less resistance per ton than 
one of tlie ^'American" type. Whether the test was strictly 
reliable or not, it certainly demonstrated that there was no 
trouble in using these heavy engines on very sharp curvature, 
and we may therefore consider that, except in the most extreme 
cases, this objection has no force whatsoever. 

* Seventh An. Rep. Am. Mast. Mech. Assn. 



§42L 



CURVATURE. 



449 



EFFECT OF CURVATURE ON OPERATING EXPENSES. 

421. Relation of radius of curvature and of degrees of 
central angle to operating expenses. The smallest consideration 
will show that the sharper the curvature the greater will be 
the tractive force required, also the greater per unit of track 
length will be the rail wear and the general w^ear and tear on 
roadbed and rolling stock. But it would be inconvenient 
to use a relation between operating expenses and radius of 
curvature, because even when such a relation was found there 
would be two elements to consider in each problem — the radius 
and the length of the curve. The method which will be here 
developed cannot claim to be strictly accurate or even strictly 
logical, but, as will be sho^^Ti later, the most uncertain elements 
of the computation have but a small influence on the final 
result, and the method is in general the only possible method of 
solution. The outline of the method is as follows: 

(1) For reasons given in detail later, it is found that the 
expenses, wear, etc., on the track from A to B will be substan- 
tially the same whether by the route M or N, The wear, etc., 




Fig. 208. 



per foot at N is of course greater, but the length of curve is 
less. Therefore the effect of the curvature depends on the 
degrees of central angle J and is independent of the radius. 



450 RAILROAD CONSTRUCTION. § 422. 

(2) At what degree of curvature is the total train resistance 
double its value on a tangent? Probably no one figure would 
be exact for all conditions. Train resistance varies with the 
velocity and with the various conditions of train loading even 
on a tangent, and it is b}^ no means certain (or even probable) 
that the ratio would be exactly the same for all conditions. 
As an average figure we may say that a train running at average 
velocity on a 10° curve will encounter a resistance due to cur- 
vature of about 10 lbs. per ton, which is the average resistance 
found on a level tangent. On a 10° curve therefore the resistance 
is doubled. 

(3) A train-mile costs about so much — approximately $1.00. 
Doubling the tractive resistance will increase certain items of 
expenditure about so much. Their combined value is so much 
per cent of the cost of a train-mile. A mile of continuous 10^ 
curve contains 528° of central angle. A mile of such track 
would add so much per cent to the average train-mile expenses, 
and each degree of central angle is responsible for -^-g of this 
increase. Since the increase is irrespective of radius and de- 
pends only on the degrees of central angle, we therefore say 
that each degree of central angle of a curve will add so much 
to the average operating expenses of a train-mile. 

The ''cost per train-mile '' considered above should be con- 
sidered as the cost of a mile of level tangent. If we for a moment 
consider that all the railroads of the country were made abso- 
lutely straight and level, it is apparent that the average cost 
per train-mile instead of being about 95c. would be somewhat 
less. The percentage should therefore be applied to this reduced 
value, but the net effect of this change would evident!}^ be 
small. 

422. Effect of curvature on maintenance of way, A 
very large proportion of the items of expense in a train -mile 
are absolutely unaffected by curvature. It will therefore 
simplify matters somewhat if we at once throw^ out all the un- 
affected items. Of the items of maintenance of way and struc- 
ture all but the first three will be thrown out. Item 4 will be 
somewhat affected when bridges or trestles occur on a curve. 
But when it is considered what a very small percentage of this 
small item (2.378%) could be ascribed to curvature, since the 
very large majority of bridges and trestles are purposely made 



§ 422. CURVATURE. 451 

straight, and since culverts, etc., are not affected, we may 
evidently ignore any variation in the item. 

Item I. Repairs of Roadway. A very large proportion of the 
sub-items are absolutely unaffected. The care of embankments 
and slopes, ditching, weeding, etc., are evidently unaffected. 
The track labor on rails and ties and the work of surfacing 
will evidently be somewhat increased and yet it is very seldom 
that the length of a track section would be decreased simply 
on account of excessive curvature. But 528° per mile is an 
excessive amount of curvature. The average for the whole 
countr}^ is about 30° per mile, and there are very few instances 
of that amount of curvature (528°) in the length of a single mile. 
As before intimated, it is reasonable to assume that the extra 
work per foot on a 20° curve would be 10 times the extra work 
on a 2° curve, which verifies the general statement that the 
extra cost varies as the degrees of central angle. Considering 
how much of this item is independent of curvature and how 
little even the track labor is affected, it is possibly overstating 
the case to allow 25% increase for 528° of curvature in one 
mile. 

Item 2. Renewals of Rails* Wellington says that some ex- 
periments made by himself and others made by Dr. Dudley 
agree in indicating that the rail wear on tangents may be con- 
sidered as 1 lb. per ^^ard per 10,000,000 tons duty, w^hile the 
extra wear on curves would be J lb. per degree of curve per 
10,000,000 tons duty. Therefore on a 10° curve the extra 
rail wear would be five times as great as on a tangent and the 
increase would therefore be 500%. On iron rails and on inferior 
steel rails the wear on tangents would be larger proportionally, 
and this is probably the reason for Wellington's adopting an 
average increase of but 300%, and this same figure will be 
adopted. 

Item 3. Renewals of Ties. Curvature affects ties by in- 
creasing the ''rail cutting'' and on account of the more frequent 
respiking, which ''spike-kills" the ties even before they have 
decayed. Wellington estimates that a tie which will last 
nine years on a tangent will last but six years on a 10° curve. 
He adds 50% for tie renewals. He considers the decrease in 
tie life to be proportional to the degree of curve and therefore 
again verifies the general statement made above regarding the 
expense of curvature » 



452 RAILROAD CONSTRUCTION. § 423. 

423. Effect of curvature on maintenance of equipment. 

Items 11, 16, 18, and 19 will be considered as unaffected. 

Item 12. Repairs and Renewals of Locomotives. Cur- 
vature affects locomotive repairs by increasing very largely the 
wear on tires and wheels, and also the wear and strain due to 
the additional power required. Wellington neglected the last 
cause since the resistance due to curvature is so small compared 
to that due to even a moderate grade. He further considered 
that only 30% of the items of engine repairs are affected at 
all by curvature, and that the effect of curvature and grades on 
these is only J or 10%, and that curvature is responsible for 
60% of that, or, finally, that only 6% of engine repairs are caused 
by curvature as it exists. He then computed that the actual 
average curvature of railroads (about 30° per mile) is but -f^ 
of the 600° (instead of 528°) in his standard mile. Therefore 
he said that 600° of curvature would increase engine repairs 
by 20X6%, or 120%. He acknowledges that the reasoning 
is not conclusive. It apparently is weak in this respect: the 
resistance, and also the wear, is less per degree of curve on the 
sharper curves than on the easier. On this account, and also 
because 528° of curvature is considered the standard, rather 
than 600°, the estimate will be cut down to 100%. (Another 
method of computation Avill be substituted for this as soon as 
possible.) 

Items 13, 14, and 15. For similar reasons the estimates for 
these items will be made 100%. The effect of curvature will 
apply to all cars about equally. 

Item 17. The repairs and renewals of shop machinery and 
tools will not be increased more than 50% per mile for the addi- 
tional repairs required of the above equipment. 

424. Effect of curvature on conducting transportation. 
We may at once throw out all items except 22, 23, 24, and 25, 
a small part of 28, and possibly 35, 36. and 37. This last group 
has already been discussed in §418; the aggregate of the three 
items is but 1.752%; ciu^^ature is responsible for only a small 
proportion of the item, and the reduction w^hich an engineer 
is able to effect would be so small that we may neglect it. 

Item 28 is somewhat analogous to the above. Curvature does 
not affect a large part of the item, but an extreme case of curva- 
ture will occasionally require an extra watchman. Consider- 
ing, however, that curvature does not in general require watch- 



§425. 



CURVATURE. 



453 



men, and that such cases are the unusual cases in mountainous 
regions where the curvature is unavoidable and not materially 
reducible, it would evidently be wrong to charge curvature in 
general ^dth such an item, although there w^ould be justifica- 
tion for it in individual cases. It ^dll therefore be ignored. 

Items 22, 23, 24, and 25. In § 407, Chapter XXI, the propor- 
tion of fuel assigned to direct hauling on a tangent is computed 
as amounting to about 55%. Since this direct resistance is as- 
sumed to be exactly doubled, w^e will charge 55% for fuel. There 
will evidently be no error worth considering in allowing the same 
proportionate amount as the charge for water, oil, waste, etc. 

'^ General expenses, '' items 47 to 53, are of course unaffected. 

425. Estimate of total effect per degree of central angle. 
Compiling the above estimates we have the following tabulation : 

TABLE XXII. EFFECT ON OPERATING EXPENSES OF CHANGES 

IN CURVATURE. 



Item 


Normal 


Per cent 


Cost per mile. 


No. 


average. 


affected. 


per cent. 


1 


10.596 


25 


2.649 


2 


1.440 


300 


4.320 


3 


3.093 


50 


1.546 


4^ 
10 f 


5.533 








20.662 




8.515 




11 


.650 








12 


5.879 


100 


5.879 


13 


2.209 


100 


2.209 


14 


6.765 


100 


6.765 


15 


.155 


100 


.155 


16 


.209 








17 


.490 


50 


.245 


18 


.040 








19 


.495 








16.892 




15.253 




20 


1.761 








21 


9.781 








22 


9.681 


55 


5.335 


23 


.671 


55 


.369 


24 


.376 


55 


.207 


25 


.184 


55 


.101 


26 (. 
46 1- 


35.340 








57.794 




6.012 




47 1 

53 r 


4.653 










100.000 




29.780 





454 



RAILROAD CONSTRUCTION. 



§425. 



According to it, 528° of curvature in one mile would increase 
the expenses of each train passing over it by 29.78% of the 
average cost of a train-mile, and according to the general prin- 
ciples laid down in § 421, 1° of central angle of any curve, no 
matter what the radius, will increase the expenses by -^^ of 
29.78%, or .0564% per degree. Therefore the cost per year 
per daily train each way is (at 95 c. per train-mile) 

95 X. 0564%) X 2X365 =39. 11 c. 

As a simple illustration (a more extended one will be given 
later), suppose that by using greater freedom •with regard to 
earthwork the crooked line sketched may be reduced to the 
simple curve shown and a curvature of, say, 110° may be re- 
duced to, say, 60°. 

Note that since the extreme tangents are identical, the sav- 
ing in central angle results from the elimination of the reversed 




Fig. 209. 

curvature and of that part of the direct curvature necessary 
to balance the reversed curvature. Assume that there are six 
daily trains each way. Then the annual saidng is 



50°X.3911X6:^$117.33, 



§ 426. CURVATURE 455 

which at 5% would justify an expenditure of $2346.60. If 
the extra cost of construction does not exceed this, the im- 
provement is justifiable, and is made all the more so if the proba- 
bilities are great that the future trafhc will largely exceed six 
trains per day. At the same time the warning regarding ^'dis- 
counting the future" with respect to expected traffic should 
not be neglected. The possible effect of change of distance 
has not been referred to in the above problem. In any case it 
is a distinct problem. According to the above sketch, the 
difference in distance is probabl}^ very slight, and consider- 
ing the compensating character of extra distance, such small 
differences may usually be disregarded. The possible effect 
of change of grade mil be discussed in the next chapter. As- 
suming that there is no difference to be considered on account 
of either grade or distance, the question hinges on the advisa- 
bility of spending !S2346.G0 for the improvement. 

426. Reliability and value of the above estimate. It should 
be realized at the outset that no extreme accuracy is claimed 
for the above estimate. The effect of curvature is somewhat 
variable as well as uncertain, but such estimates have this great 
value. Vary the estimates of indi\ddual items as you please 
(within reason), and the final result is still about the same and 
may be used to guide the judgment. As an illustration, sup- 
pose that the item of renewals of rails is assumed to be affected 
400% rather than 300%, the justifiable expenditure to avoid 
the curvature in the above case may similarly be computed 
as $2460, an increase of less than 5%. But, after all, the real 
question is not whether the improvement is worth $2346 or 
$2460. The extra work involved may perhaps be done for $500 
or it may require $10000. The above general method furnishes 
a criterion which, while not accurate, is so much better than a 
reliance on vague judgment that it should not be ignored. 



COMPENSATION FOR CURVATURE. 

427. Reasons for compensation. The effect of curvature on 
a grade is to increase the resistance by an amount which is equiv- 
alent to a material addition to that grade. On minor grades 
the addition is of little importance, but when the grade is nearly 
or quite the ruling grade of the road, then the additional resist- 
ance induced by a curve will make that curve a place of maxi- 



456 



RAILROAD CONSTRUCTION. 



§427. 



mum resistance and the real maximum will be a 'Sqrtual grade" 
somewhat higher than the nominal maximum. If, in Fig. 210, 



Tang. 




Fig. 210. 

AN represents an actual uniform grade consisting of tangents 
and curves, the ^Sdrtual grade" on curves at BC and DE may 
be represented by BC and DE. If BC and DE are very long, 
or if a stop becomes necessary on the curve, then the full dis- 
advantage of the curve becomes developed. If the whole grade 
may be operated without stoppage, then, as elaborated further 
in the next chapter, the w^hole grade may be operated as if equal 
to the average grade, AF, which is better than BC, although 
much worse than AN. The process of '^compensation" con- 
sists in reducing the grade on every curve by such an amount 
that the actual resistance on each curve, due to both curvature 
and grade, shall precisely equal the resistance on the tangent. 
The practical effect of such reduction is that the 'Sirtual" grade 
is kept constant, while the nominal grade fluctuates. 

One effect of this is that (see Fig. 211) instead of accomplish- 




FiG. 211. 



ing the vertical rise from A to G (i.e., HG) in the horizontal 
distance AH, it requires the horizontal distance AK. Such an 
addition to the horizontal distance can usually be obtained by 
proper development, and it should always be done on a ruling 



§ 428. CURVATURE. 457 

grade. Of course it is possible that it will cost more to accom- 
plish this than it is worth, but the engineer should be sure of 
this before allowing this virtual increase of the grade. 

European engineers early realized the significance of unre- 
duced curvature and the folly of laying out a uniform ruUng 
grade regardless of the curvature encountered. Curve compen- 
sation is now quite generally allowed for in this country, but 
thousands of miles have been laid out without any compensa- 
tion. A very common limitation of curvature and grade has 
been the alliterative figures 6° curvature and 60 feet per mile 
of grade, either singly or in combination. Assuming that the 
resistance on a 6° curve is equiA^alent to a 0.3% grade (15.84 feet 
per mile), then a 6° curve occurring on a 60-foot grade would 
develop more resistance than a 75-foot grade on a tangent. 
The '^mountain cut-off'' of the Lehigh Valley Railroad near 
Wilkesbarre is a fine example of a heavy grade compensated 
for curvature, and yet so laid out that the virtual grade is uni- 
form from bottom to top, a distance of several miles. 

428. The proper rate of compensation. This evidently is the 
rate of grade of Vv^hich the resistance just equals the resistance 
due to the curve. But such resistance is variable. It is greater 
as the velocity is lov/er; it is generally about 2 lbs. per ton 
(equivalent to a 0.1% grade) per degree of curve when starting 
a train. On this account, the compensation for a curve which 
occurs at a knov/n stopping-place for the heaviest trains should 
be 0.1% per degree of curvCo The resistance is not even strictly 
proportional to the degree of curvature, although it is usually 
considered to ]:>e so. In fact most formulae for curve resistance 
are based on such a relation. But if the experimentally deter- 
mined resistances for lov/ curvatures are applied to the excessive 
curvature of the Nev^ York Elevated road, for example, the 
rules become ridiculous. On this account the compensation 
per degree of curve may be made less on a sharp curve than on 
an easy curve. The compensation actually required for very 
fast trains is less than for slow trains, sa}^ 0.02 or 0.03% per 
degree of curve; but since the comparatively slow and heaA^y 
freight trains are the trains w^hich are chiefly limited by ruling 
grade, the compensation must be made with respect to those 
trains. From 0.04 to 0.05% per degree is the rate of compen- 
sation most usually employed for average conditions. Curves 
which occur below a known stopping-place for all trains need 



458 RAILROAD CONSTRUCTION. § 429. 

not be compensated, for the extra resistance of the curve will 
be simply utilized in place of brakes to stop the train. If a curve 
occurs just above a stopping-place, it is very serious and should 
be amply compensated. Of course the down-grade traffic need 
not be considered. 

It sometimes happens that the ordinary rate of compensa- 
tion w^ill consume so much of the vertical height (especially if 
the curvature is excessive) that a steeper through grade must 
be adopted than was first computed, and then the trains might 
stall on the tangents rather than on the curves. In such cases 
a slight reduction in the rate of compensation might be justi- 
fiable. The proper rate of compensation can therefore be 
estimated from the following rules : 

(1) On the upper side of a stopping-place for the heaviest 
trains compensate 0.10% per degree of curve. 

(2) On the lower side of such a stopping-place do not com- 
pensate at all. 

(3) Ordinarily compensate about 0.05% per degree of curve. 

(4) Reduce this rate to 0.04% or even 0.03% per degree 
of curve if the grade on tangents must be increased to reach 
the required summit. 

(5) Reduce the rate somewhat for curvature above 8° or 10°. 

(6) Curves on minor grades need not be compensated. 

429. The limitations of maximum curvature. What is the 
maximum degree of curvature which should be allowed on any 
road? The true answer is probably that there is no definite 
limit. It has been shown that sharp curvature does not prevent 
the use of the heaviest types of engines, and although a sharp 
curve unquestionably increases operating expenses, the increase 
is but one of degree with hardly any definite limit. The general 
character of the country and the gross capital available (or 
the probable earnings) are generally the true criterions. 

A portion of the road from Denver to Leadville, Col., is an 
example of the necessity of considering sharp curvature. The 
traffic that might be expected on the line was so meagre and 
yet the general character of the country was so forbidding 
that a road built according to the usual standards would have 
cost very much more than the traffic could possibly pay for. 
The line as adopted cost about $20,000 per mile, and yet in a 
stretch of 11.2 miles there are about 127 curves. One is a 25° 
20' curve, tw enty-four are 24° curves, twenty-five are 20° curves, 



1 



§ 429. CURVATURE. 459 

and seventy-two are sharper than 10°. If 10° had been made 
the Umit (a rather high hmit according to usual ideas), it is 
probable that the line would have been found impracticable 
(except with prohibitive grades) unless four or five times as 
much per mile had been spent on it, and this would have ruined 
the project financially. 

For mam'^ years the main-line traffic of the Baltimore and 
Ohio H. R. has passed over a 300-foot curve (19° 10') and a 
400-foot curve (14° 22') at Harper's Ferry. A few^ years ago 
some reduction w^as made in this by means of a tunnel, but 
the fact that such a road thought it wise to construct and operate 
such cur^'es (and such illustrations on the heaviest-traffic roads 
are quite common) show^s how foolish it is for an engineer to 
sacrifice money or (w^hich is much more common) sacrifice 
gradients in order to reduce the rate of curvature on a road 
w^hich at its best is but a second- or third-class road. 

Of course such belittling of the effects of curvature may 
be (and sometimes is) carried to an extreme and cause an engi- 
neer to fail to give to curvature its due consideration. Degrees 
of central angle should always be reduced by all the ingenuity 
of the engineer, and should only be limited by the general rela- 
tion betw^een the financial and topographical conditions of the 
problem. Easy curvature is in general better than sharp curva- 
ture and should be adopted when it may be done at a small 
financial sacrifice, especially since it reduces distance generally 
and may even cut down the initial cost of that section of the 
road. But large financial expenditures are rarely, if ever, jus- 
tifiable wiiere the net result is a mere increase in radius without 
a reduction in central angle. An anal3^sis of the changes which 
have been so extensively made during late years on the Penn. 
R. R. and the N. Y., N. H. & H. R. R. wiU show^ invariably a 
reduction of distance, or of central angle, or both, and perhaps 
incidentally an increase in radius of curvature. There are but 
few, if any, cases W'here the sole object to be attained by the 
improvement is a mere increase in radius, 



CHAPTER XXIII. 

GRADE. 

430. Two distinct effects of grade. The effects of grade on 
train expenses are of two distinct kinds; one possible effect is 
very costly and should be limited even at considerable expen- 
diture; the other is of comparatively little importance, its cost 
being slight. As long as the length of the train is not limited, 
the occurrence of a grade on a road simply means that the engine 
is required to develop so many foot-pounds of work in raising 
the train so many feet of vertical height. For example, if a 
freight train weighing 600 tons (1,200,000 lbs.) climbs a hill 
50 feet high, the engine performs an additional work of creating 
60,000,000 foot-pounds of potential energy. If this height is 
surmounted in 2 miles and in 6 minutes of actual time (20 
miles per hour), the extra work is 10,000,000 foot-pounds per 
minute, or about 303 horse-power. But the disadvantages of 
such a rise are always largely compensated. Except for the fact 
that one terminus of a road is generally higher than the other, 
every up grade is followed, more or less directly, by a down grade 
which is operated partly by the potential energy acquired during 
the previous climb. But when we consider the trains running 
in both directions even the difference of elevation of the termini 
is largely neutralized. If we could eliminate frictional resist- 
ances and particularly the use of brakes, the net effect of minor 
grades on the operation of minor grades in both directions would 
be zero. Whatever was lost on any up grade would be regained 
on a succeeding down grade, or at any rate on the return trip. 
On the very lowest grades (the limits of which are defined later) 
we may consider this to be hterally true, viz., that nothing is 
lost by their presence; whatever is temporarily lost in climbing 
them is either immediately regained on a subsequent light down 
grade or is regained on the return trip. If a stop is required 
at the bottom of a sag, there is a net and uncompensated loss 
of energy. 

460 



§ 431. GRADE. 461 

On the other hand, if the length of trains is Hmited by the 
grade, it will require more trains to handle a given traffic . The 
receipts from the traffic are a definite sum. The cost of hand- 
ling it will be nearly in proportion to the number of trains. 
Anticipating a more complete discussion, it may be said as an 
example that increasing the ruling grade from 1.20% (63.36 
feet per mile) to 1.55% (81.84 feet per mile — an increase of 
about 18.5 feet per mile) will be sufficient to increase the re- 
quired number of trains for a given gross traffic about 25%, 
i.e., five trains will be required to handle the traffic which four 
trains would have handled before at a cost slightly more than 
four-fifths as much. The effect of this on di\adends may readil}'- 
be imagined. 

431. Application to the movement of trains of the laws of 
accelerated motion. When a train starts from rest and acquires 
its normal velocity, it overcomes not only the usual tangent 
resistances (and perhaps curve and grade resistances), but it 
also performs work in storing into the train a vast fund of kinetic 
energy. This work is not lost, for every foot-pound of such 
energ}^ may later be utilized in overcoming resistances, pro- 
vided it is not wasted by the action of train-brakes. If for a 
moment we consider that a train runs without any friction, 
then, when running at a velocity of v feet per second, it possesses 
a kinetic energy which would raise it to a height h feet, when 

h = — , in which g is the acceleration of gravity =32.16. Assum- 
ing that the engine is exerting just enough energy to overcome 
the frictional resistances, the train would chmb a grade until the 
train was raised h feet above the point where its velocity was v. 
When it had climbed a height h' (less than h) it would ha^^e a 
velocit}^ %\=\/2g{h — h'). As a numerical illustration, assume 

i; = 30 miles per hour = 44 feet per second. Then /i = 7— = 30. 1 feet , 

2g ' 

and assuming that the engine was exerting just enough force 

to overcome the rolling resistances on a level, the kinetic 

energy in the train would carry it for two miles up a grade of 

15 feet per mile, or half a mile up a grade of 60 feet per mile. 

When the train had climbed 20 feet, there w^ould still be 10.1 

feet left and its velocity would be ri=\/2g(10.1) =25.49 feet 
per second = 17.4 miles per hour. These figures, however, must 
be slightly modified on account of the weight and the rcA^olving 



462 RAILROAD CONSTRUCTION. § 432. 

action of the wheels, which form a considerable percentage 
of the total weight of the train. When train velocity is beiag 
acquired, part of the work done is spent in imparting the energy 
of rotation to the driving-wheels and various truck-wheels of 
the train. Since these wheels run on the rails and must turn 
as the train moves, their rotative kinetic energy is just as effec- 
tive — as far as it goes — in becoming transformed back into 
useful work. The proportion of this energy to the total kinetic 
energy has already been demonstrated (see Chapter XVI, 
§ 347). The value of this correction is variable, but an average 
value of 5% has been adopted for use in the accompanying 
tabular form (Table XXIII), in which is given the corrected 
*S^elocity head" corresponding to various velocities in miles 
per hour. The table is computed from the following formula : 

. -yMn ft. per sec. 1.4667tun m. perh. ^^^^^^2 
Velocity head= ^^^ = ^^ ^^ =0m344v^ 

adding 5% for the rotative kinetic energy of the wheels, 0.00167r^ 



The corrected velocity head therefore equals 0.03511 r^ 

Part of the figures of Table XXIII were obtained by inter- 
polation and the final hundredth may be in error by one unit, 
but it may readil}^ be shoT\Ti that the final hundredth is of no 
practicable importance. It is also true that the chief use made 
of this table is with velocities much less than 50 miles per hour. 
Corresponding figures may be obtained for higher velocities, if 
desiyed, by multiplying the figure for half the velocity by fou7\ 

432. Construction of a virtual profile. The following simple 
demonstration will be made on the basis that the ordinar}^ 
tractive resistances and also the tractive force of the locomo- 
tive are independent of velocity. For a considerable range of 
velocity which includes the most common freight-train velocities 
this assumption is so nearly correct that the method will give 
an approximately correct result, but for higher velocities and 
for more accurate results a more complicated method (given 
later) must be used. The following demonstration will serve 
w^ell as a preliminary to the more accurate method. It ma}^ 
best be illustrated by considering a simple numerical example. 

Assuming that a train is passing A (see Fig. 212), running at 
30 miles per hour. Assume that the throttle is not changed or 
any brakes applied, but that the engine continues to exert the 



§43: 



GRADE. 



463 



TABLE XXIII. VELOCITY HEAD (REPRESENTING THE KINETIC 

energy) of trains MOVING AT VARIOUS VELOCITIES. 



Vel. 






















mi. 
hr. 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


10 


3.51 


3.58 


3.65 


3.72 


3.79 


3.87 


3.95 


4.02 


4.10 


4.17 


11 


4.25 


4.33 


4.41 


4.49 


4.57 


4.65 


4.73 


4.81 


4.89 


4.97 


12 


5.06 


5.15 


5.23 


5.32 


5.41 


5.50 


5.58 


5.67 


5.75 


5.84 


13 


5.93 


6.02 


6.12 


6.21 


6.31 


6.40 


6.50 


6.59 


6.69 


6.78 


14 


6.88 


6.98 


7.08 


7.19 


7.29 


7.39 


7.49 


7.60 


7.70 


7.80 


15 


7.90 


8.00 


8.11 


8.22 


8.33 


8.44 


8.55 


8.66 


8.77 


8.88 


16 


8.99 


9.10 


9.21 


9.32 


9.43 


9.55 


9.67 


9.79 


9.91 


10.03 


17 


10.15 


10.2/ 


10.39 


10.51 


10.63 


10.75 


10.87 


10.99 


11.12 


11.25 


18 


11.38 


11.50 


11.63 


11.76 


11.89 


12.02 


12.15 


12.28 


12.41 


12.55 


19 


12.68 


12.81 


12.95 


13.08 


13.22 


13.35 


13.49 


13.63 


13.77 


13.91 


20 


14.05 i 14.19 


14.33 


14.47 


14.61 


14.75 


14.89 


15.04 


15.19 


15.34 


21 ' 15.49 i 15.64 


15.79 


15.94 


16.09 


16.24 


16.39 


16.54 


16.69 


16.84 


22 ; 17.00 


17.15 


1 7.30 


17.46 


17.62 


17.78 


17.94 


18.10 


18.26 


18.42 


23 , 18.58 


18.74 


18.90 


19.06 


19.22 


19.38 


19.55 


19.72 


19.89 


20.06 


24 1 20.23 


20.40 


20.57 


20.74 


20.91 


21.08 


21.25 


21.42 


21.59 


21.77 


25 ' 21.95 


22.12 


22.30 


22.48 


22.66 


22.84 


23.02 


23.20 


23.38 


23.56 


26 i 23.74 


23.92 


24.10 


24.28 


24.46 


24.65 


24.84 


25.03 


25 22 


25.41 


27 i 25.60 


25.79 


25.98 


26.17 


26.36 


26.55 


26.74 


26.93 


27.13 


27.33 


28 ■ 27.53 


27.73 


27.93 


28.13 


28.33 


28.53 


28.73 


28.93 


29.13 


29.33 


29 1 29.53 


29.73 


29.93 


30.13 


30.34 


30.55 


30.76 


30.97 


31.18 


31.39 


30 1 31.60 


31.81 


32.02 


32.23 


32.44 


32.65 


32.86 


33.08 


33.30 


33.52 


31 33.74 


33.96 


34.18 


34.40 


34.62 


34.84 


35.06 


35.28 


35.50 


35.72 


32 35.95 


36.17 


36.39 


36.62 


36.85 


37.08 


37.31 


37.54 


37.77 


38.00 


33 ; 38.23 


38.46 


38.69 


38.92 


39.15 


39.38 


39.62 


39.86 


40.10 


40.34 


34 1 40.58 


40.82 


41.06 


41.30 


41.54 


41.78 


42.02 


42.26 


42.51 


42.76 


35 43.01 


43.26 


43.51 


43.76 


44.01 


44.26 


44.51 


44.76 


45.01 


45.26 


36 45.51 


45.76 


46.01 


46.26 


46.52 


46.78 


47.04 


47.30 


47.56 


47.82 


37 ! 48.08 


48.34 


48.60 


48.86 


49.12 


49.38 


49.64 


49.91 


50.18 


50.45 


38 : 50.72 


50.99 


51.26 


51.53 


51.80 


52.07 


52.34 


52.61 


52.88 


53.15 


39 i 53.42 


53.69 


53.96 


54.23 


54.51 


54.79 


55.07 


55.35 


55.63 


55.91 


40 


56.19 


56.47 


56.75 


57.03 


57.31 


57.59 


57.87 


58.16 


58.45 


58.74 


41 


59.03 


59.32 


59.61 


59.90 


60.19 


60.48 


60.77 


61.06 


61.35 


61.64 


42 


61.94 


62.23 


62.52 


62.82 


63.12 


63.42 


63.72 


64.02 


64.32 


64.62 


43 


64.92 


65.22 


65.52 


65.82 


66.12 


66.43 


66.74 


67.05 


67.36 


67.67 


44 


67.98 


68.29 


68.60 


68.91 


69.22 


69.53 


69.84 


70.15 


70.46 


70.78 


45 71.10 


71.42 


71.74 


72.06 


72.38 


72.70 


73.02 


73.34 


73.66 


73.98 


46 


74.30 


74.62 


74.94 


75.26 


75.59 


75.92 


76.25 


76.58 


76.91 


77.24 


47 


77.57 


77.90 


78.23 


78.56 


78.89 


79.22 


79.55 


79.89 


80.23 


80.57 


48 


80.91 


81.25 


81.59 


81.93 


82.27 


82.61 


82.95 


83 29 


83.63 


83.97 


49 


84.32 


84.66 


85.00 


85.34 


85.69 


86.04 


86.39 


86.74 


87.09 


87.44 


50 


87.79 


88.14 


88.49 


88.85 


89.20 


89.55 


89.91 


90.26 


90.61 


90.97 



same draw-bar pull. At A its ^S^elocity head" is that due to 30 
miles per hour, or 31.60 feet. At B it has gained 40 feet more, 
and its velocity is that due to a velocity head of 71.60 feet, or 
slightly over 45 miles per hour. At B' its velocity is again 30 
miles per hour and velocity head 31.60 feet. At C the velocity 
head is but 6.60 feet and the velocity about 13.7 miles per hour. 



464 



RAILROAD CONSTRUCTION 



§432. 




As the train runs from Cto D its velocity increases to 30 miles at 
C and to over 45 miles per hour at D. At E the velocity again 
becomes 30 miles per hour. Although there will be some slight 
modifications of the above figures in actual practice, yet the above 
is not a fanciful theoretical sketch. Thousands of just such 
undulations of grade are daily operated in such a way, without 
disturbing the throttle or applying brakes, and the draw-bar 
pull, if measured by a dynamometer, would be found to be 
practically constant. Of course the above ease assumes that 



Actual profile 



Fig. 212. 
there are no stoppages and that the speed through the sags is 
not so great that safety requires the application of brakes. 
Observe that the ''virtual profile" is here a straight line — as it 
alwa3^s is when the draw-bar pull is constant. The virtual 
profile (in this case as well as in ever}^ other case, illustrations 
of which will follow) is found by adding to the actual profile 
at any point an ordinate which represents the ''velocity head" 
due to the velocity of the train at that point. 

As another case, assume that a train is climbing the grade AE 
and exerting a pull just sufficient to maintain a constant velocity 

^ / up that grade. Then A'B' (parallel 
' to A 5) is the virtual profile, A A* 
representing the velocity head. A 
stop being required at (7, steam is 
shut off and brakes are applied 
at 5, and the velocity head BB' 
reduces to zero at C. The train 
starts from C, and at D attains a velocity corresponding to 
the ordinate DU . At D the throttle may be slightly closed 
so that the velocity will be uniform and the virtual grade is 
D'E', parallel to DE. 

From the above it may be seen that a virtual profile has the 
following properties : 

(a) When the velocity is uniform, the virtual profile is parallel 
with the actual. 




Fig. 213. 



§ 433. GRADE. 465 

(b) When the velocity is increasing the profiles are separating ; 
when decreasing the profiles are approaching. 

(c) When the velocity is zero the profiles coincide. 

433. Use, value, and possible misuse. The essential feature 
respecting grades is the demand on the locomotive. From the 
foregoing it msLj readily be seen that the ruling grade of a road 
is not necessarily the steepest nominal grade. When a grade 
may be operated by momentum, i.e., when every train has an 
opportunity to take ^'a run at the hill/' it may become a very 
harmless grade and not limit the length of trains, while another 
grade, actually much less, which occurs at a stopping-place 
for the heaviest trains, will require such extra exertion to get 
trains started that it may be the worst place on the road. There- 
fore the true way to consider the value of the grade at any criti- 
cal place on the road is to construct a virtual profile for that 
section of the road. The required length of such a profile is 
variable, but in general may be said to be limited by points on 
each side of the critical section at which the velocit}^ is definite, 
as at a stopping-place (velocity zero), or a long heavy grade where 
it is the minimum permissible, say 10 or 15 miles per hour. 

Since the velocities of different trains vary, each train will 
have its own virtual profile at any particular place. The fast 
passenger trains are generall}^ unaffected, practically. The 
requirement of high average speed necessitates the use of power^ 
ful engines, and grades wliich would stall a heavy freight mil 
only cause a momentary and harmless reduction of speed of 
the fast passenger train. 

A possible misuse of virtual profiles lies in the chance that a 
station or railroad grade crossing may be subsequently located 
on a heavy grade that was designed to be operated by momen- 
tum. But this should not be used as an argument against the 
employment of a virtual profile. The virtual profile shows the 
actual state of the case and only points out the necessity, if an 
unexpected requirement for a full stoppage of trains at a critical 
point has developed, of changing the location (if a station), or 
of changing the grade by regrading or by using an overhead 
crossing. Examples of such modifications are given in Chap- 
ter XXIV, The Improvement of Old Lines. 

434. Undulatory grades. Advantages. ^loney can generally 
he saved by adopting an actual profile which is not strictly 
.uniform — the matter of compensation for curvature being here 



46C RAILROAD CONSTRUCTION. § 434. 

ignored. Its effect on the operation of trains is harmless pro- 
vided the sag or hump is not too great. In Fig. 214 the undu- 
latory grade may actually be operated as a uniform grade AG. 
The sag at C must be considered as a sag, even though BC is actu- 
ally an up grade. But the engine is supposed to be working 

F g^ ^rrrrr/^' 




V^ 



Fig. 214. 
hard enough to carry a train at uniform velocity up a grade AG. 
Therefore it gains in velocity from B to C, and from C to D loses 
an equal amount. It may even be proven that the time re- 
quired to pass the sag will be slightly less than the time required 
to run the uniform grade. 

Disadvantages. The hump at F is dangerous in that, if the 
velocity at E is not equal to that corresponding to the extra 
velocity-head ordinate at F, the train will be stalled before 
reaching F. In practice there should be considerable margin. 
Any train should have a velocity of at least 10 miles per hour 
in passing any summit. This corresponds to a velocity head 
of 3.51 feet. An extra heavy head wind, slippery rails, etc., 
may use up any smaller margin and stall the train. If the 
grade AG is a ruling grade, then no hump should be allowed 
under any circurnstances. For the heaviest trains are supposed 
to be so made up that the engine w^ill just haul them up the 
ruling grades — of course with some margin for safety. Any 
increase of this grade, however short, would probably stall the 
train. 

Safe limits. It is quite possible to have a sag so deep that 
it is not safe to allow freight trains to rush through them with- 
out the use of brakes. The use of brakes of course adds a 
distinct element of cost. To illustrate: If a freight train is 
running at a velocity of 20 miles per hour (velocity head 14.05 
feet) and encounters a sag of 25 feet, the velocity head at the 
bottom of the sag will be 39.05 feet, which corresponds to a 
velocity of 33.3 miles per hour. This approaches the limnt of 
safe speed for freight trains, and certainly passes the limit for 
trains not equipped with air-brakes and automatic couplers. 



§ 435. GRADE. 467 

The term ^'safe limits '^ as used here, refers to the Umits within 
which a freight train may be safely operated without the appli- 
cation of brakes or varying the work of the engine. Of course 
much greater undulations are frequently necessary and are 
safely operated, but it should be remembered that they add a 
distinct element to the cost of operating trains and that they 
must not be considered as harmless or that they should be 
introduced unless really necessary. 



MINOR GRADES. 

435. Basis of cost of minor grades. The basis of the com- 
putation of this least objectionable form of grade is as follows: 
The resistance encountered by a train on a level straight track 
is somewhat variable, depending on the velocity and the num- 
ber and character of the cars, but for average velocities we 
may consider that 10 lbs. per ton is a reasonable figure. This 
value agrees fairly well ^^ith the results of some dynamometer 
tests made by Mr. P. H. Dudley, using a passenger train of 
313 tons running at about 50 miles per hour. It also agrees 
T\ath Searles's formulae (leased on experiments) for the resist- 
ance of a freight train with 40 cars running 25 miles per hour. 
Ten pounds per ton is the grade resistance of a 0.5% grade, or 
that of 26.4 feet per mile. On the above basis, a 0.5% grade 
will just double the tractive resistance on a level straight track. 
We may compute, as in the previous chapter, the cost of doubling 
the tractive resistance for one mile. But since the extra resist- 
ance is due to lifting the train through 26.4 feet of elevation, 
we may dlA^ide the extra cost of a mile of 0.5% grade by 26.4 
and we have the cost of one foot of difference of elevation, and 
then (disregarding the limiting effect of grades) we may say that 
this cost of one foot of difference of elevation will be independent 
of the rate of grade. There are, however, limitations to this 
general proposition which will be developed in the next section. 

436. Classification of minor grades. These are classified with 
reference to their effect on the operation of trains. In the first 
class are grades which may be operated without changing the 
work of the engine and which have practically no other effect 
than a harmless fluctuation of the velocity. But a grade which 
belongs to this class when considering a fast passenger train will 
belong to another class when considering a slow but heavy 



^^r 




468 RAILROAD CONSTRUCTION. § 436. 

freight train. And since it is the slow heavy freight trains 
which must be chiefly considered, a grade will usually be classi- 
fied with respect to them. The limit of class A (the harmless 

class) therefore depends on the 
n' .,^''^ maximum allowable speed. The 

effect of a sag on speed will depend 
T^^-.~>^__^^X on the vertical feet of drop rather 

B than on the rate of grade, for 

with the engine working as usual 

on even a light down grade a train 
would soon exceed permissible speed. Assume that a freight 
train runs at an average speed of 15 miles per hour vdth a 
minimum of 10 miles and a permissible maximum of 30 miles 
per hour. Assume that a train runs up the grade at A with 
a uniform velocity of 15 miles per hour, i.e., the engine is 
working so that the velocity would be uniform to C. How 
much sag (BB') can there be without the speed exceeding 30 
miles per hour? 

Velocity head for 30 miles per hour 31 . 60 

'' "Id " " '' 7.90 

The drop BB' vnW therefore be 23.70 

While each case must be figured by itself, considering the 
probable velocity of approach and the maximum permissible 
velocity, we may say that a sag of about 24 feet will ordinarily 
mark the limit of this class. With a higher velocity of approach 
even this limit will be much reduced. 

The classification therefore applies to sags and humps and 
to the vertical feet of drop or climb which are involved, rather 
than to grade per se. The practical application of these prin- 
ciples is necessarily confined to humps or sags which are pos- 
sibly removable and does not apply to the long grades which 
are essential to connect predetermined points of the route — 
grades w^hich are irreducible except by development and which 
must be studied as ruling grades (see §§ 440-445). 

The application therefore consists in the comparative study 
of two proposed grades, noting the relative energy required 
to operate them and the probable cost. The depth in feet 
saved would be the maximum difference between the grades, 
and the classification will depend on the necessary method of 
operating the trains. 



§ 437. GRADE. 469 

,. The next classification (B) applies to drops so deep that 
|i steam must be shut off when descending the grade, while the 
|! work required of the engine when ascending the opposite grade 
|i is correspondingly increased. The loss is not so serious as 
in the next case, but the inability of the engine to work con- 
tinuously may result in a failure to accumulate sufficient kinetic 
energy to carry the train over a succeeding summit. 

The third class (C) includes the grades so long that brakes 
must be applied to prevent excessive velocity. The loss in- 
volved is very heavy; the brakes require power for their appli- 
cation, they wear the brake-shoes and wheel-tires, they destroy 
kinetic or potential energy which had previously been created, 
while the tax on the locomotive on the corresponding ascend- 
ing grade is very great. The ascending grade may or may not 
be a ruling grade. 

437. Effect on operating expenses. As in Chapter XXII 
we may at once throw out a large proportion of the items of 
expense of an average train-mile. In " maintenance of way 
and structures" items 4 to 10 are evidently unaffected. 

Item I. Repairs of Roadway. It is very plain that a 
large proportion of the sub-items are al^solutely unaffected 
by minor grades. In fact it is a little difficult to ascribe any 
definite increase to any sub-item. The rail wear is somewhat 
increased and this will have some effect on the trackwork, 
but on the other hand the increased grade sometimes results 
in better drainage and therefore less work to keep the track 
in condition. Wellington allows 5% increase as a 'liberal 
estimate'' for class C, and no increase for the other classes. 

Item 2. Renewals of Rails. Observations of rail wear 
on heavy grades show that it is much greater than on level 
tangents. But usually such heavy grades are operated by 
shorter trains or mth the help of pusher engines, and the pro- 
portion of engine tonnage to the total is much greater than is 
ordinarily the case And since an engine has much greater 
effect on rail wear than cars, particularly on account of the 
use of sand, an excess of engine tonnage would have a marked 
effect. But such circumstances would inevitably accompany 
ruling grades and not minor grades. Nevertheless the effect 
of the use of sand on up grades and the possible skidding of 
wheels on down grades will wear the rails somewhat. Even 
the possible slipping, of the drivers, although sand is not used, 



470 RAILROAD CONSTRUCTION. § 438. 

will wear the rails. Wellington allows 10% increase for class 
C and 5% for class B. 

Item 3. Renewals of Ties. The added wear of ties might 
be considered proportional to that of the rails except that, as 
in the case of the roadbed in general, the better drainage secured 
by the grade will tend to increase the life of the ties. Welling- 
ton makes the estimate the same as for item 1, 5% for class C 
and no increase for the other classes. 

Maintenance of equipment. Items 11, 16, 17, 18, and 19 
are evidently unaffected. Items 12 to 15. The chief sub- 
items of increase will evidently be the repairs and renewals of 
wheels and brake-shoes both for locomotives and cars. In 
the case of cars the draw-bar is apt to suffer from severe alter- 
nate compression and extension due to push and pull. The 
locomotive mechanism will suffer somewhat from the extra 
demands on it, and the boiler on account of the intermittent 
character of the demands on it. It would seem as if such 
effects would be quite large, but an examination of the com- 
parative records of engine and car repairs on mountain divisions 
and on comparatively level divisions shows no such difference 
as might be expected. On this account Wellington cuts down 
these items to 4% for class C and 1% for class B. 

Conducting transportation. As in Chapter XXI, § 407, since 
the resistance is assumed to be doubled, we may take the same 
figure (55%) as the cost of the fuel for climbing the 26.4 feet. 
But the total cost of both the rise and fall is to be considered. 
In class B, although steam^ is shut off, heat (and fuel) is wasted 
by mere radiation. This has been estimated (Chapter XXI, 
§ 407) as about 5%. Therefore we ma}^ allow 60% for class B, 
For class C we must allov/ in addition the energy spent in 
applying brakes, which we may assume as 5% more, making 
65%. Items 23, 24, and 25 may be estimated similarly. The 
other items under this head as well as General Expenses are 
evidently unaffected. 

438. Estimate of the cost of one foot of change of elevation. 
Collecting these estimates, we have the accompanying tabular 
form, showing that the percentage of increase for operating 
grades of class B or class C will be 6.77% and 8.56%, respect- 
ively. On the basis of an average cost of 95 c. per train-mile, 
the additional cost for the 26.4 feet in one mile would be 6.43 c. 
and 8.13 c, or 0.24 c. and 0.31 c. per foot. For each train per 



§ 439. GRADE. 471 

day each way per year the value per foot of difference of ele- 
vation is: 

For class 5: 2X365X$0.0024=$1.75; 
'' '' C: 2 X 365 X $0.0031 =$2.26. 

It will frequently happen that a grade must be considered a^ 
belonging to class C for heavy freight trains, and that it belongs 
to class B or even class A for other trains. If no Sunday trains 
are run, 313 should be used instead of 365 as a multipher in 
the above equations. 

439- Operating value of the removal of a hump in a grade. 
As a simple illustration of the above, suppose that the irregular 
grade A BCD may be cut down to the uniform grade AD, either 
by direct lowering of the track or by a modification of align- 
ment. If a freight train, running to the left, passes C at a veloc- 
ity of 15 miles per hour (velocity head, 7.90 feet) and drops 



Fig. 216. 

down to B (a vertical fall of 62.40 feet) without shutting off 
steam, its velocity head at B would be 70.30 feet and its velocity 
44.8 miles per hour, which is inadmissible. Therefore the hump 
certainly does not belong to class A, for freight trains. Sup- 
pose that steam is shut off when passing C. Then, if Ave con- 
sider the average value of 0.5% as the grade which is equivalent 
to the normal rolling resistances, we may consider the 1.3% 
grade as an 0.8% grade, down which the train passes wdthout 
resistance. An 0.8% grade for 4800 feet would be a drop of 
38.40 feet, and the velocity head at B would be 7.90 + 38.40 = 
46.30 feet, and the velocity would be 36.3 miles per hour, which 
may or may not be considered as inadmissible for freight trains. 
According to the decision the grade belongs to class C or to 
class B. For trains moving to the right, or up grade, there is no 
definite criterion, but a grade of 1.3% is a severe tax on a loco- 
motive, even if it is not a ruling grade. Ignoring its possible 



472 



RAILROAD CONSTRUCT- ; ; . , 



§ 139. 



TABLE XXIV. EFFECT ON OPERATING EXPENSES OF CHANGES 

IN GRADE. 





Item (abbreviated).* 


Normal 
average. 


Class B. 


Class C. 


No. 


Per cent j Cost 
affected 'per mile 


Per cent 
affected 


Cost 
per mile 


1 


Roadway 


10.596 
1.440 
3.093 
5.533 



5 





.07 
.07 




5 

10 

5 




53 


2 


Rails 


14 


3 


Ties 


15 


4-10 


Bridges, buildings, etc . 
Maintenance of way. 







20.662 




.07 




.82 


11 
12 
13 
14 
15 
16 
17 


Superintendence 

Repairs locomotives . . 
Repairs pass, cars .... 
Repairs freight cars . . . 
Repairs work cars .... 
Marine equipment .... 
Shops 


.650 

5.879 

2.209 

6.765 

.155 

.209 

.490 

.040 

.495 




1 
1 
1 
1 








.06 
.02 

.07 
.00 







4 
4 
4 
4 







.23 
.09 

.27 

.06 






18 
19 


*Stat. and printing .... 
Other expenses 

Main, of equip 









16.892 




.15 




.65 


20 
21 


Superintendence 

Enginemen 


1.761 

9.781 

9.681 

.671 

.376 

.184 

35 . 340 




60 
60 
60 
60 








5.81 

.40 

.23 

.11 






65 
65 
65 
65 








22 


Fuel. . 


6.29 


23 


Water. .. . 


.44 


24 


Oil, etc. . . . 


.24 


25 
26-46 


Other supplies 

Train service, station 
service, etc 


.12 





Conducting transp . . 






57.794 




6.55 




7.09 


47-53 


General expenses 


4.653 


















100.000 




6.77 




8.56 



* For full title of item see Table XX. 

limiting effect (which is a separate matter), the value of the 
33.6 feet is evidently 

33.6 X$2.26 =$75.94 per daily train for class C 



and 



B. 



33.6X$1.75=$58.80 '' '' '' " ' 

Assuming that there are six daily trains each way for which the 
grade would be classified as C, and four others which could 
operate the grade as a ^' ^" grade, the total annual cost would be 

6X$75.94 =$455.64 
4 X$58.80 = $235.20 



$690.84 



§ 440. GRADE. 473 

This annual cost, capitalized at 5%, equals $13817, which 
is the jvistifiable expenditure to avoid the hump. Assuming 
that the cut would involve 300000 culoic yards, at 20 c. por 
cubic yard, it would cost $60000 to make the through cut. On 
the above basis the cut would not be justifiable, but a small 
part of such cutting would so reduce the hump that it would 
not belong to class C for any trains, and it might even be re- 
duced to class A. Of course other solutions are possible. A 
slightly different route may be chosen from B to D, involving 
a different distance, different curvature, and a marked reduc- 
tion in the hump. The effect of all such changes must be com- 
bined and their net effect determined 



RULING GRADES. 

440. Definition. Ruling grades are those which limit the 
weight of the train of cars which may be hauled by one engine. 
The subject of '' pusher grades" will be considered later. For 
the present it will suffice to say that on all well-designed roads 
the large majority of the grades on any one division are kept 
below some limit which is considered the ruling grade. If a 
heavier grade is absolutely necessary no special expense will 
be made to keep it below a rate where the resistance is twice 
(or possibly three times) the resistance on the ruling grade, and 
then the trains can be hauled unbroken up these few special 
grades with the help of one (or two) pusher engines. So far 
as limitation of train length is concerned, these pusher grades 
are no worse than the regular ruling grades and, except for the 
expense of operating the pusher engines (which is a separate 
matter), they are not appreciably more' expensive than any 
ruling grade. As before stated, the engineer cannot alter very 
greatly the general level of the road when the general route has 
been decided on. He may remove sags or humps, or he may 
lower the natural grade of the route by development in order 
to bring the grade within the adopted limit of ruling grade. 
The financial value of removing sags and humps has been con- 
sidered. It now remains to determine the financial relation 
between the lowest permissible ruling grade and the money 
which may profitably be spent to secure it. 

441. Choice of ruling grade. It is of course impracticable for 
an engine to drop off or pick up cars according to the grades 



474 RAILROAD CONSTRUCTION. § 442. 

which may be encountered along the line. A train load is made 
up at one terminus of a division and must run to the other 
terminus. Excluding from consideration any short but steep 
grades which ma}'- alwaijs be operated by momentum, and also 
all pusher grades, the maximum grade on that division is the 
ruling grade. 

It will evidently be economy to reduce the few grades which 
naturally would be a little higher than the great majority of 
others until such a large amount of grade is at some uniform 
limit that a reduction at all these places would cost more than 
it is worth. The precise determination of this limit is prac- 
tically impossible, but an approximate value may be at once 
determined from a general survey of the route. The distance 
apart of the termini of the division into their difference of ele- 
vation is a first trial figure for the rate of the grade. If a grade 
even approximately uniform is impossible owing to the eleva- 
tions of predetermined intermediate points, the worst place 
may be selected and the natural grade of that part of the route 
determined. If this grade is much steeper than the general 
run of the natural grades, it may be policy to reduce it by devel- 
opment or to boldh^ plan to operate that place as a pusher 
grade. The choice of possible grades thus has large hmita- 
tions, and it justifies A'^ery close study to determine the best 
combination of grades and pusher grades. When the choice 
has narrowed down to two limits^ the lower of which may be 
obtained by the expenditure of a definite extra sum, the choice 
may be readily computed, as will be developed. 

442. Maximum train load on any grade. The tractive power 
of a locomotive has been discussed in Chap. XV, § 322. The 
net train load which -may be placed behind any engine is the 
difference between the weight of the engine itself and the gross 
load which can be handled under the given circumstances, with 
a given weight on the drivers. Since the design of locomotives 
is so variable, it is impracticable to show in tabular form the 
power of all kinds of locomotiA^es on all grades. In Table XXV 
are given the tractive powers of locomotives of a wide range of 
types and weights and with various ratios of adhesion. They 
may be accepted as typical figures and will serve to compute the 
effect of variations of grade on train load. In Table XXVI is 
given the total train resistance in pounds per ton for various grades 
and for various values of track resistances. By a combination 



§442, 



GRADE, 



47 



TABLE XXV. TRACTIVE POWER OF VARIOUS TYPES OF LOCO- 
MOTIVES AT VARIOUS RATES OF ADHESION. 



Kind. 


Gauge. 


Total weight 

of engine and 

tender. 


Weight 
of en- 
gine 
only. 


Weight 

on 
drivers. 


Tractive power when 
rate of adhesion is 




lbs. 


tons. 


i- 


ia 


i 


American. . . 
Mogul 


Nar. 


49,000 
80,000 
81,000 


24.5 

40 

40.5 


32,000 

49,000 
51,000 


22,000 
32,000 
42,000 


5,500 

8.000 

10,500 


4,950 
7,200 
9,450 


4.400 
6,400 
8,400 


10-wheel. . . 

Consol 

it 


( ( 


87,000 

62,000 

144,000 


43.5 
31 

72 


55,000 
39,000 
94,000 


42,000 
34,000 
84,000 


10,500 

8,500 

21.000 


9,450 

7,650 
18,900 


8,400 

6.800 

16,800 


American.. . 
*'Chautau".. 
Mogul 


Stand. 


104,000 
314,600 
206,000 


52 
157.3 
103 


62,000 
190,600 
126,000 


40,000 

99,400 

106,400 


10,000 
24,850 
26,600 


9,000 
22.305 
23,940 


8,000 

19.880 
21,280 


10-wheel . . . 
Co-sol 


- 


276,000 
214,000 
324,800 


138 
107 
162.4 


176,510 
120.0;)0 
204,800 


127,010 
106,000 
181,200 


31.752 
26.500 
45,300 


28,577 
23,850 
40,770 


25,402 
21,200 
36,240 



of these two tables the net train load on any grade under given 
conditions may be quickly determined For example, an 
ordinary consolidation engine having a weight of 106000 
pounds on the drivers (see Table XXV) A\dll have a tractive 
force of 26500 pounds under fair conditions of track, when the 
adhesion ratio is j. When climbing slowly up a grade of 1.30% 
the tractive resistance will be about 32 pounds per ton if the roll- 
ing-stock and track are fair — assuming a tractive resistance on 
a level of 6 pounds per ton. Dividing 26500 by 32 we have 
828 tons, the gross train load. Subtracting 107 tons, the weight 
of the engine and tender in working order, we have 721 tons, 
the net load. Incidentally we may note that, cutting do^\Ti 
the grade to 0.90% (a reduction of only 21.12 feet per mile), 
the resistance per ton is reduced to 24 pounds and the gross 
train load is increased to 1104 tons and the net load to 997 
tons — an increase of about 38%. 

As another numerical example, consider a contractor's loco- 
motive (not referred to in Table XXV), a light four-wheel-con- 
nected-tank narrow-gauge engine, ^^-ith a total weight of 12000 
pounds, all on the drivers. On the rough temporary track 
used by contractors the tractive ratio may be as low as -J. 
The tractive adhesion should therefore be taker as 2400 pounds. 
Assume that the grade when hauhng "empties" is 4.7% and 



476 RAILROAD CONSTRUCTION. § 443. 

that the tractive resistance on such a track on a level is 10 pounds 
per ton. E}^ Table XXVI, the total train resistance is therefore 
(by interpolation) 104 pounds per ton. 2400-f-104 =23 tons; 
subtracting the weight of the engine we have 17 tons, the net 
load of empty cars — perhaps twenty cars weighing 1700 pounds 
per car. 

In general, and to compute accurately the train load under 
conditions not exactly given in the tables, the maximum train 
load may be computed according to the following rule : 

The maximum load behind an engine on any grade may be 
found by multiplying the weight on the drivers by the ratio of 
adhesion and dividing this by the sumi of the grade and tractive 
resiGtances per ton; this giA^es the gross load, from which the 
weight of the engine and tender must be subtracted to find the 
net load. 

443. Proportion of the traffic a^ected by the ruling grade. 
Some very light trafhc roads are not so fortunate as to have 
a traffic which vill be largeh^ affected by the rate of the ruling 
grade. When passenger trafBc is light, and when, for the sake 
of encouraging traffic, more frequent trains are run than are 
required from the standpoint of engine capacity, it may happen 
that no passenger trains are really limited by any grade on the 
road — i.e., an extra passenger car could be added if needed. 
The maximum grade then has no worse eiTect (for passenger 
trains) than to cause a harmless reduction of speed at a few points. 
The local freight business is frequently aff3cted in practically 
the same way. All coal, mineral, or timber roads are affected 
by the rate of ruling grade as far as such traffic is concerned. 
Likewise the through business in general merchandise, especially 
of the heavy traffic roads, will generally be affected by the rate 
of ruHng grade. Therefore in computing the effect of ruling 
grade, the total number of trains on the road should not ordi- 
narily be considered, but only the trains to which cars are added, 
until the limit of the hauling power of the engine on the ruling 
grades is reached. 

444. Financial value of increasing the train load. The gross 
receipts for transporting a given amount of freight is a definite 
sum regardless of the number of train loads. The cost of a 
train mile is practicall}^ constant. If it were exactly so, the 
saving in operating expenses would be strictly proportional 
to the number of trains saved. How will the cost per trairv 



i 



§444. 



GRADE. 



477 



table xxyi. total train resistance per ton (of 2000 

pounds) on various grades. 







When tractive 


re- 


Grade. 


sistance on a level 






in pounds per ton is 


Rate 


Feet 












per 


per 


6 


7 


8 


9 


10 


cent. 


mile. 












0.00 


0.00 


6 


7 


8 


9 


10 


.05 


2.64 


7 


8 


9 


10 


11 


.10 


5.28 


8 


9 


10 


11 


12 


.15 


7.92 


9 


10 


11 


12 


13 


.20 


10.56 


10 


11 


12 


13 


14 


0.25 


13.20 


11 


12 


13 


14 
15 


15 
16 


.30 


15.84 


12 


13 


14 


.35 


18.48 


13 


14 


15 


16 


17 


.40 


21.12 


14 


15 


16 


17 


18 


.45 


23.76 


15 


16 


17 


18 


19 


0.50 


26.40 


16 


17 

18 


18 
19 


19 
20 


20 

21 i 


.55 


29.04 


17 


.60 


31.68 


18 


19 


20 


21 


22 1 


.65 


34.32 


19 


20 


21 


22 


23' 


.70 


36.96 


20 


21 


92 


23 


24! 


0.75 


39.60 


21 


22 


23 


24 
25 


26 


.80 


42.24 


22 


23 


24 


.85 


44.88 


23 


24 


25 


26 


27 


.90 


47.52 


24 


25 


26 


27 


28 


0.95 


50.16 


25 


26 


27 


28 


29 


1.00 


52.80 


26 


27 


28 


29 


30 


.05 


55.44 


27 


28 


29 


30 


31 ! 


.10 


58.08 


28 


29 


30 


31 


32 1 


.15 


60.72 


29 


30 


31 


32 


33 


.20 


63.36 


30 


31 


32 


33 


34 


1.25 


66.00 


31 


32 


33 


34 


35 


.30 


68.64 


32 


33 


34 


35 


36 


.35 


71.28 


33 


34 


35 


36 


37 


.40 


73.92 


34 


35 


36 


37 


38 


.45 


76.56 


35 


36 


37 


38 


39 


1.50 


79.20 


36 


37 


38 


39 


40 


.55 


81.84 


37 


38 


39 


40 


41 


.60 


84.48 


38 


39 


40 


41 


42 


.65 


87.12 


39 


40 


41 


42 


43 


.70 


89.76 


40 


41 


42 


43 


44 


1.75 


92.40 


41 


42 


43 


44 


45 


.80 


95.04 


42 


43 


44 


45 


46 


.85 


97.68 


43 


44 


45 


46 


47 


.90 


100.32 


44 


45 


46 


47 


48 


1.95 


102.96 


45 


46 


47 


48 


49 


2.00 


105.60 


46 


47 


48 


49 


50 







When tractive 


re- 


Grade. 


sistance on a level 






in pounds per ton is 


Rate 


Feet 












per 


per 


6 


7 


8 


9 


10 


cent. 


mile. 












2.00 


105.60 


46 


47 


48 


49 


5) 


.05 


108.24 


47 


48 


49 


50 


5] 


.10 


110.88 


48 


49 


50 


^] 


52 


.15 


113.52 


49 


50 


51 


52 


53 


.20 


116.16 


50 


51 


52 


53 


54 


2.25 


118.80 


51 


52 


53 


54 


55 


.30 


121.44 


52 


53 


54 


55 


5o 


.35 


124.08 


53 


54 


55 


56 


57 


.40 


126.72 


54 


55 


56 


57 


58 


.45 


129.36 


55 


56 


57 


58 


59 


2.50 


132,00 


56 


57 


58 


59 


60 


.55 


134.64 


57 


58 


59 


60 


61 


.60 


137.28 


58 


59 


60 


61 


62 


.65 


139.92 


59 


60 


61 


62 


63 


.70 


142 . 56 


60 


61 


62 


63 


64 


2.75 


145.20 


61 


62 


63 


64 


65 


.80 


'147.84 


62 


63 


64 


65 


66 


.85 


150.48 


63 


64 


65 


66 


67 


.90 


153.12 


64 


65 


66 


67 


68 


.95 


155.76 


65 


66 


67 


68 


60 


3.00 


158.40 


66 


67 


68 


69 


70 


.05 


161.04 


67 


68 


69 


7C 


71 


.10 


163.68 


68 


69 


70 


71 


72 


.15 


166.32 


69 


70 


71 


72 


73 


.20 


168.96 


70 


71 


72 


73 


74 


3.25 


171.60 


71 


72 


73 


74 


75 


.30 


174.24 


72 


73 


74 


75 


76 


.35 


176.88 


73 


74 


75 


76 


77 


.40 


179.52 


74 


75 


76 


77 


78 


.45 


182.16 


75 


76 


77 


78 


79 


3.50 


184.80 


76 


77 


78 


79 


80 


4.00 


211.20 


86 


87 


88 


89 


90 


4.50 


237.60 


96 


97 


98 


99 


100 


5.00 


264.00 


106 


107 


108 


109 


110 


5.50 


290.40 


116 


117 


118 


119 


120 


6.00 


316.80 


126 


127 


128 


129 
139 


130 


6.50 


343 . 20 


136 


137 


138 


140 


7.00 


369.60 


146 


147 


148 


149 


150 


8.00 


422.40 


166 


167 


168 


169 


170 


i 9.00 


475.20 


186 


187 


188 


189 


190 


10.00 


528.00 


206 


207 


208 


209 


210 



mile vary when by a reduction in ruling grade more cars are 
handled in one train than before ? First, compute the effect 



478 . RAILROAD CONSTRUCTION. § 444, 

of increasing the train load so that one less engine will handle 
the traffic, or, for example, that an engine can haul 11 cars 
instead of 10 or 44 instead of 40 — that 10 engines \Yil\ do the 
work for which 11 engines would be required with the steeper 
grade. What will be the relative cost of running 10 heavy 
trains rather than 11 lighter trains, or, rather, what will be the 
extra cost of the extra engine ? 

Since the gross traffic to be handled is assumed to be the 
same, the number of cars required to handle it will also bf^ the 
same whatever the number of trains, and the effect of those 
cars on the vrear and tear of track, etc., will evidently be constant. 
The locomotive, on account of the greater concentration of 
loading of the driver wheels, damages the track (in proportion 
to its tonnage) much more than the cars. It has been estimated 
that the locomotive is responsible for one half of the track 
wear Such an estimate is verified by the wear of rails on steep 
tracks around coal-mines where standard cars are hauled by 
cables. If we assume that 50% of Items 2 and 3 and of that 
part of Item 1 which varies with tonnage is due to the locomo- 
tives, then the extra expense caused by the extra engine will 
be 50% of Items 2 and 3 and 50%^ of 25% of Item 1. The 
other items of maintenance of way are unaffected e?:cept that 
truss bridges, trestles, and the maintenance of a few buildings 
will be slightly affected by the extra locomotive. But the 
actual effect is quite indefinite and is evidently very small. 

Maintenance of equipment: Engine repairs will evidently be 
affected according to the mileage. Throughout the ruling 
grade of the road (by whichever system of grades) the engines 
(assumed of uniform style) are w^orking at their utmost capacity. 
On the lighter grades and level sections the engines wall have 
easier work when the cars are fewer and this will have a tendency 
to reduce engine repairs. Suppose that by decreasing the 
number of cars 10% on the easy grades the engine repairs on 
each engine are reduced 2%. There is little or no justification 
for estimating the reduction to be more than this. Then on 
the ten engines the saving is 20% of the average charge for 1 
engine. Suppose that by decreasing the number of cars 20%, 
on the easy grades the engine repairs are reduced 4%, on the 
five engines they are reduced 20% again. In either case the 
net added cost due to the extra engine would be but 80% of 
the average cost While the above estimate is bul^ a guess, 



§ 445. GPtADE. 479 

yet it is ver^^ evident that the extra cost for this item is but 
Httle less than the normal charge. 

Car repairs will be reduced by a decrease in the number of 
cars per train. The average draw-bar pull ^\dll be less, the 
wear and tear due to stoppage and starting will be less. This 
is the one item in which an increased number of trains for the 
same tonnage is an actual advantage. The saving per car is 
evidently greater when 4 trains are increased to 5 than when 
10 trains are increased to 11; but the saving per train added 
on is constant. Wellington estimates the saving to be 10%. 
His basis of calculation is somewhat different, but it reduces 
to the same thing. The estimate applies chiefly to Item 14 
and to Item 13 in so far as passenger trains are affected by 
ruling grade. The other items of maintenance of equipment are 
but little, if any, affected. 

Conducting transportation. Items 20, 21, 26, 27, 28, 29, 30, 
31, 32, 34, 35, 36, 37, 45, and 46 may be considered as varying 
according to the train mileage. While some of them seem to 
have but little direct connection with train mileage, yet if a 
road increases its traffic from 10 trains a day to 20 trains a day 
all of these items seem to increase in due proportion. 

Fuel, etc., for locomotives (Items 22-25) will increase nearly 
as the engine mileage. In either case the engines w^ork to the 
limit of their capacity on the ruling grades. In either case the 
loss of heat due to radiation is the same. But the engines with 
the lighter trains work a httle easier on the hght or level grades. 
By the same course of reasoning as was given regarding engine 
repairs the fuel saving from the normal requirement for the 
extra engine will be about the same no matter whether there 
is an addition of one engine in 5 or 10. The saving in fuel will 
be assumed at 25% of the normal consumption, or rather that 
the use of the extra engine adds 75% of the normal charge for 
fueL The same estimate applies to items 23, 24, and 25. 

Car mileage is unaffected. Items 38 to 44 will be considered 
as unaffected, also the general expenses. 

445. Operating value of a reduction in the rate of the ruling 
grade. Collecting the above estimates, we have Table XXVII. 
To this must be added something for the capital cost of the extra 
engine. Assume that it costs $10000 and that its mileage life 
is 800000 miles. This makes an average charge of 1.25 c. per 
mile. Of course the cost of operation, maintenance, and repairs 



480 



RAILROAD CONSTRUCTION. 



§ 445. 



is included in the tabulated expense. 54.82% of 95 c. =52.08 c. 
Adding 1.25 c., we have 53.33 c. 



TABLE XXVII. COST OF AN ADDITIONAL TRAIN TO HANDLE 

A GIVEN TRAFFIC. 



No. 


Item (abbreviated). 


Normal 
average. 


Per cent 
affected. 


Cost per 
cent. 


1 


Roadway 


10.596 
1.440 
3.093 
5.533 


12.5 
50 
50 



1.32 


2 


Rails 


.72 


3 


Ties 


1.55 


4-10 


Bridges, buildings, etc 







Maintenance of way 


20.662 




3.59 








11 


Superintendence 


.650 
5.879 
2.209 
6.765 
1.389 



80 

5 
10 







12 

13 

14 

15-19 


Repairs of locomotives 

Repairs of passenger-cars 

Repairs of freight-cars 

Miscellaneous 


4.70 
.11 
.68 












Maintenance of equipment .... 


16.892 




3.91 








20 


Superintendence 


1.761 

9.781 

10.912 

24.285 
2.094 
2.085 
5.646 
1.229 


100 

100 

75 

100 



100 



100 


1.76 


21 


Enginemen 


9.78 


22-25 


Fuel, etc 


8.18 


26-32 


Train service, etc 


24.28 


33 


Car mileage 





34-37 


Damages, etc 


2.09 


38-44 


Miscellaneous 





45-46 


Stationery, etc 


1.23 










Conducting transportation. . . . 


57.793 




47.32 








47-53 


General expenses 


4.653 










100.000 




54.82 









As a practical application of the above figures, assume that 
on a constructed and operated road the ruling grade on a 100- 
mile division is 1.6%; the actual traffic affected by ruling grade 
is 8 daily trains with a net load of 552 tons or 4416 tons. It 
is found that with an expenditure of $400000 the ruling grade 
may be reduced to 1.2%. Will it pay? At 1.2% grade the net 
load behind an 80-ton consolidation engine, with 48 tons on 
the drivers, adhesion J, is 720 tons. The traffic (4416 tons) 
may therefore be hauled by 6 engines, the balance, less than 
100 tons, being taken care of by lighter trains not affected by 
the ruling grade. Since the additional cost of the engine drawing 
lighter trains is 53 c. per mile, the saving by reducing from 
8 engines to 6 is that due to 2 engines. The annual saving 
is therefore 2 X $0.5333x100x365 =$38930.90, which capital- 



§ 446. GRADE. 481 

ized at 5% =$778618. This shows that if the improvement 
can be accompHshed for $400000 it is worth while. 

As in other similar problems, it must be reiterated that al- 
though there are some more or less uncertain elements in the 
above estimates, yet with a considerable margin for error in 
individual items the value of the whole improvement will not 
be very greatly altered and the estimate will be infinitely better 
than an indefinite reliance on vague ''judgment." Of course 
certain items in the above estimates are somewhat variable 
and should be altered to fit the particular case to be computed. 

/ 

PUSHER GRADES. 

446. General principles underlying the use of pusher engines. 

On nearly all roads there are some grades which are greatly 
in excess of the general average rate of grade and these heavy 
grades cannot usually be materially reduced without an ex- 
penditure which is excessive and beyond the financial capacity 
of the road= If no pusher engines are used, the length of all 
heavy trains is limited by these grades. The financial value 
of the reduction of such ruling grades has already been shown. 
But in the operation of pusher grades there is incurred the 
additional cost of pusher-engine service, for a pusher engine 
must run tirice over the grade for each train which is assisted. 
It is possible for this additional expense to equal or even exceed 
the advantage to be gained,. In any case it means the adoption 
of the lesser of two evils, or the adoption of the more economical 
method. A simple examxple will illustrate the point. Assume 
that at one point on the road there is a grade of 1.9% which 
is five miles long. Assume that all other grades are less than 
0.92%. I*^ pushers are not to be used the net capacit}^ of a 
107-ton consolidation engine with 53 tons on the drivers, assum^ 
ing /o adhesion and 6 pounds per ton for normal resistance, 
will be 435 tons, and that will be the maximum weight of train 
allowable. By using pusher engines on this one 5-mile grade 
the train load is at once doubled and the number of trains 
cut down one half. This double load, 870 tons, can easily be 
hauled by one engine up the 92% grades. As a rough com- 
parison, free from details and allowances, we may say: 

(a) 10 trains per day over a 100-mile division, 435 tons 
net per train, will require 1000 engine miles daily. 



482 HAILROAD CONSTRUCTION. § 447 

(b) 5 trains per day handling the same traffic, 870 tons 
net per train, with 2X5X5 pusher-engine mileSj will require 
(5X 100) -h (2X5X5) =550 engine miles daily. There is thus 
a large saving in the number of engine miles and also in the 
number of the engines required foi: the w^ork. Moreover, the 
engines are working to the limit of their capacity for a much 
larger proportion of the time, and their work is therefore m.ore 
economically done. The work of overcoming the normal 
resistances of so many loaded cars over so many miles of track 
and of lifting so many tons up the gross differences of elevation 
of predetermined points of the line is approximately the same 
whatever the exact route, and if the grades are so made that 
fewer engines working more constantly can accomplish the 
work as well as more engines which are not hard worked for a 
considerable proportion of the tim.e, the economy is very ap- 
parent and unquestionable. Wellington expresses it concisely: 
" It is a truth of the first im.portance that the objection to 
high gradients is not the work which the engines have to do 
on them, but it is the work which they do not do when they 
thunder over the track with a light train behind them, from 
end to end of a division, in order that the needed power maj' 
be at hand at a few scattered points where alone it is needed.^' 

447. Balance of grades for pusher service. In the above 
illustration the "through" grade and the "pusher" grade are 
"balanced" for the use of one equal pusher. It is therefore 
evident that if some intermediate grade (such as 1 A%) were 
permitted, it could only be operated by (a) making it the ruling 
grade and cutting doYxn all train loads from 870 tojis to 594 tons, 
or (h) operating it as a pusher grade, although with a loss of 
economy, since two engines would have much more power than 
necessary. The proper plan i'n such a case would be to strive 
to reduce the 1-4% grade to 0.92%, or, if that seemed imprac- 
ticable, to attempt to get an operating advantage at the expense 
of an increase of the 1.4% grade to anything short of 1.9%. 
For the increase in rate of grade would cost almost nothing, and 
some advantage might be obtained Avhich would practically 
compensate for the introduction of a pusher grade. Another 
possible solution would be to operate the 1 9% with two pushers, 
adopt a corresponding grade for use with one pusher and a 
corresponding ruling grade for through trains. With the above 



§447. 



GKADJE. 



483 



data these three grades would be 1.90% 1.27%, and 0.54%, 
obtained as follows: 

TractiA'e power of three engines = 106000 X -io X S = 71550 
pounds. 

Resistance on 1.9% grade= 6 + (20X1.9) =44 lbs. per ton. 

71550^-44= 1626 = gross load in tons. 

1626 -(3X107) =1305 = net load in tons. 

1305 4- (2X107) =1519=gross load on the one-pusher grade. 

Tractive power of two engines = 47700 lbs. 

47700 4-1519= 31.40= possible tractive force in lbs. per ton. 

(31.40 — 6) -^ 20= 1.27% = permissible grade for one pusher. 

1305-h 107= 1412 =gross load on the through grade. 

Tractive power of one engine =23850 lbs. 

23350^1412 =16.89 = possible tractive force in lbs. per ton. 

(16.89-6^ --20 =0.54% = permissible through grade. 

It should be realized that, assuming the accuracy of the 
normal resistance (6 lbs.) and the normal adhesion (/(j) and 
with the use of 107-ton locomotives with 53 tons on the drivers, 
the above figures are precisely what is required for hauling 
with one, two, and three engines. Other types of engines, other 
values for resistance and adhesion will vary considerably the 
gross load in tons which may be hauled up those grades, but 
starting with 0.54% as a through grade, the corresponding 
values for one and for two pushers would vary but slightly 
from those given. To show the tendency of these variations, 
the corresponding values have been computed as follows: 



Adhesion. 


Resistance 
per ton. 


Load on 
drivers. 


Through 
grade. 


One-pusher 
grade. 


Two-pusher 
grade. 


i 


6 lbs. 

7 " 
6 " 

6 " 

7 •' 


53 tons. 
53 " 
53 " 
53 " 
53 " 


0.54% 

.54%, 

.54% 
.54% 
.54% 


1.27% 
1.31% 
1.28% 
1.26% 
1.29% 


1.90% 
1.96% 
1.93% 
1.86% 
1.92% 



The above form shows that increasing the resistance per ton 
and decreasing the adhesion have opposite effects on altering 
the ratio of these grades, and as a storm, for example, would 
increase the resistance and decrease the adhesion, the changes 
in the raiio would be compensating although the absolute 
reduction in train load might be considerable. 



484 RAILROAD CONSTRUCTION. § 448. 

In Table XXVIII is shown a series of "balanced" grades on 
which a given net train load may be operated b}^ means of one 
or two pusher engines. For example, assuming a track resistance 
of 6 pounds per ton, a consolidation engine of the type shown 
in the table can haul a train weighing 977 tons (exclusive of 
the engine) up a grade of 0.80%. If this is the maximum 
through grade, pusher grades as high as 1.70% for one pusher, 
or 2.46% for two pushers, may be introduced and the same 
net load may be hauled up these grades. 

The ratios of pusher grade to through grade, as given in 
Table XXVIII, are exactly true only for the conditions named 
as to weight and type of engine, ratio of adhesion, and norma 
track resistance. But a little comparative study of the two 
halves of Table XXVIII and of the tabular form given on page 
483 will show that although the net load which can be hauled 
on any grade varies considerably with the normal track re- 
sistance and also with the ratio of adhesion, yet the ratios of 
through io pusher grade, for either one or two pushers, varies 
but slightly with ordinary changes in these conditions. There- 
fore when the precise conditions are unknown or variable, the 
figures of Table XXVIII may be considered as applicable to 
any ordinary practice, especially for preliminary computations. 
For final calculations on any proposed ruling grade and pusher 
grade, the whole problem should be worked out on the principles 
outlined above and on the basis of the best data obtainable. 

Problem: If the through ruling grade for the road has been 
established at 1.12%, what pusher grades are permissible? 
Answer: Interpolating in Table XXVIII, we may employ a 
grade of 2.22% if the track and road-bed are to be such that a 
tractive resistance of 6 pounds per ton can be expected. With 
a poorer track, the normal resistance assumed as 8 pounds per 
ton, the rate is raised to 2.27%. The increase in rate of pusher 
grade ^vith increase of resistance is due to the fact that the 
net load hauled is less — so much less that on the pusher grade 
a larger part of the adhesion is available to overcome a grade 
resistance. 

448. Operation of pusher engines. The maximum efficiency 
in operating pusher engines is obtained when the pusher engine 
is kept constantly at work, and this is facilitated when the pusher 
grade is as long as possible, i.e., when the heavy grades and the 
great bulk of the difference of elevation to be surmounted is 



§448. 



GRADE. 



485 



TABLE XXVIII. BALANCED GRADES FOR ONE, TWO, AND 

THREE ENGINES. 
Basis. — Through and pusher engines alike; consolidation type; total 
weight, 107 tons; weight on drivers, 53 tons; adhesion, ^^y, giving a trac- 
tive force for each engine of 23850 lbs.; normal track resistance; 6 (also 8) 
lbs. per ton. 





Track resistance, 


6 lbs. 


Track resistance, 


8 lbs. 






Corresponding 




Corresponding 


Through 


Net load 


pusher g 


rade for 


Net load 


pusher grade for 


grade. 


for one 


same net load. 


for one 


same net load. 




engine m 






engine m 








tons (2000 






tons (2000 








lbs.). 


One 


Two 


lbs.). 


One 


Two 






pusher. 


pushers. 




pusher. 


pushers. 


Level. 


3868 tons 


0.28% 


0.55% 


2874 tons 


0.37% 


0.72% 


0.10% 


2874 ' ' 


0.47% 


0.82% 


2278 • ' 


0.56% 


0.98% 


0.20% 


2278 " 


0.66% 


1.08% 


1880 • ' 


0.74% 


1.2.3% 


0.30% 


1880 •' 


0.84% 


1.33% 


1596 " 


0.92% 


1.47% 


0.40% 


1596 " 


1.02% 


1.57% 


1384 " 


1.09% 


1.70% 


0.50% 


1384 " 


1.19% 


1.80% 


1218 " 


1.27% 


1.92% 


0.60% 


1218 " 


1.37% 


2.02% 


1085 " 


1.44% 


2.14% 


0.70% 


1085 " 


1.54% 


2.24% 


977 " 


1.60% 


2.36% 


0.80% 


977 " 


1.70% 


2.46% 


887 " 


1.77% 


2.56% 


0.90% 


887 " 


1.87% 


2.66% 


810 " 


1.93% 


2.76% 


1.00% 


810 " 


2.03% 


2.86% 


745 " 


2.09% 


2.96% 


1 . 10% 


745 " 


2.19% 


3.06% 


688 " 


2.24% 


3.15% 


1.20% 


688 " 


2.34% 


3.25% 


638 " 


2.40% 


3.33% 


1.30% 


638 " 


2.50% 


3.43% 


594 *' 


2.55% 


3.51% 


1.40% 


594 " 


2.65% 


3.61% 


555 " 


2.70% 


3.68% 


1.50% 


555 " 


2.80% 


3.78% 


521 " 


2.85% 


3.85% 


1.60% 


521 " 


2.95% 


3.95% 


489 " 


2.99% 


4.02% 


1.70% 


489 " 


3.09% 


4.12% 


461 •' 


3.13% 


4.17% 


1.80% 


461 " 


3.23% 


4.27% 


435 " 


3.27% 


4.33% 


1.90% 


435 " 


3.37% 


4.43% 


411 " 


3.42% 


4.49% 


2.00% 


411 " 


3.52% 


4.59% 


350 ' ' 


3.55% 


4.63% 


2.10% 


390 " 


3.65% 


4.73% 


370 " 


3.68% 


4.78% 


2.20% 


370 " 


3.78% 


4.88% 


352 " 


3.81% 


4.92% 


2.30% 


352 " 


3.91% 


5.02% 


335 " 


3.94% 


5.05% 


2.40% 


335 ". 


4.04% 


5.15% 


319 " 


4.07% 


5.19% 


2.50% 


319 " 


4.17% 


5.29% 


304 " 


4.20% 


5.32% 



at one place. For example, a pusher grade of three miles fol- 
lowed by a comparatively level stretch of three miles and then 
by another pusher grade of two miles cannot all be operated as 
cheaply as a continuous pusher grade of five miles. Either 
the two grades must be operated as a continuous grade of eight 
miles (sixteen pusher miles per trip) or else as two short pusher 
grades, in which case there would be a very great loss of time 
and a difficulty in so arranging the schedules that a train need 



486 RAILROAD CONSTRUCTION. § 449. 

not wait for a pusher or the pushers need not waste too much 
time in idleness waiting for trains. If the level stretch were 
imperative, the two grades would probably be operated as one, 
but an effort should be made to bring the grade^:^ together. It 
is not necessary to bring the trains to a stop to uncouple the 
pusher engine, but a stop is generally made for coupling on, and 
the actual cost in loss of energy and in wear and tear of stopping 
and starting a heavy train is as great as the cost of running 
an engine light for several miles. 

There are two ways in which it is possible to economize in 
the use of pusher engines, (a) When the traffic of a road is 
so very light that a pusher engine will not be kept reasonably 
busy on the pusher grade it may be worth while to place a 
siding long enough for the longest trains both at top and bottom 
of the pusher grade and then take up the train in sections. 
Perhaps the worst objection to this method is the time lost 
while the engine runs the extra mileage, but with such very 
light traffic roads a little time more or less is of small consequence. 
On light traffic roads this method of surmounting a heavy grade 
w^ill be occasionally adopted even if pushers are never used. 
If the traffic is fluctuating, the method has the advantage 
of onl}^ requiring such operation when it is needed and avoiding 
the purchase and operation of a pusher engine which has but 
little to do and which might be idle for a considerable proportion 
of the year, (h) The second possible method of economizing 
is only practicable when a pusher grade begins or ends at or 
near a station yard where switching-engines are required. In 
such cases there is a possible economy in utilizing the switching- 
engines as pushers, especially when the work in each class is 
small, and thus obtain a greater useful mileage. But such cases 
are special and generally imply small traffic. 

A telegraph-station at top and bottom of a pusher grade is 
generally indispensable to effective and safe operation. 

449. Length of a pusher grade. The virtual length of the 
pusher grade, as indicated by the mileage of the pusher engine, 
is always somewhat in excess of the true length of the grade 
as shown on the profile, and sometimes the excess length is 
very great. If a station is located on a lower grade within a 
mile or so of the top or bottom of a pusher grade, it will ordina- 
rily be advisable to couple or uncouple at or near the station, 
since the telegraph-station, switching, and signaling may be 



§ 450. GRADE. 487 

more economically operated at a regular station. If the extra 
engine is coupled on ahead of the through engine (as is some- 
times required by law for passenger trains) the uncoupling at 
the top of the grade may be accom.plished by running the assist- 
ant engine ahead at greater speed after it is uncoupled, and, 
after running it on a siding, clearing the track for the train. 
But this requires considerable extra track at the top of the grade. 
Therefore, when estimating the length of the pusher grade, 
the most desirable position for the terminal sidings must be 
studied and the length determined accordingly rather than 
by measuring the mere length of the grade on the profile. Of 
course these odd distances are alvrays excess; the coupling or 
uncoupling should not be done while on the grade. 

450. Cost of pusher-engine service. The cost evidently de- 
pends partly on the mileage run, while some items are wholly 
independent of the mileage. A pusher engine, when working 
on grades where the conditions are fairly favorable, will ac- 
complish a mileage of 100 to 125 miles per day, and this is 
about equal to that of an ordinary freight engine. Therefore 
such items as wages which are independent of mileage will be 
assumed to cost as maich per mile as they do for ordinary train 
ser\dce. If the mileage is less than this, an extra allowance 
should be made. 

The effect of a pusher engine on maintenance of way may 
be considered to be the same as that produced by an additional 
engine, as developed in § 444. The same allowance (3.59%) 
will therefore be made. The cost of repairs and renewals of 
locomotives may be estimated the same as for other engines. 
Wages for engine and round-house men will be the same. There 
is certainly no ground for considering that the cost of fuel and 
other engine supplies can be materially less than the usual 
figures. On the return trip down the grade the engine runs 
almost ^\ithout steam (after getting started), but, on the other 
hand, the engine works hard when climbing up the grade. The 
cost of switchmen, etc., and telegraph expenses (Items 28 and 
29) will evidently add their full quota. Collecting these items, 
we have 36.27% or 31.46 c. for each mile run. Adding, as in 
§445, 1.25 c. as interest charge on the cost of the engine, we 
have 35.71 c. Then each mile of the incline will cost twice 
this or 71.42 c. for a round trip, or 71.42x365 =$260 per year 
per mile of incline per daily train needing assistance. 



488 



RAILROAD CONSTRUCTION. 



§451, 



TABLE XXIX. ITEMS OF THE COST PER MILE OF A PUSHER ENGINE. 



No. 


Items. 


Normal 
average. 


Per cent 
affected. 


Cost per 

engine 

mile. 

per cent. 


1 


Repairs of roadway 


10.596 
1.440 
3.093 
5.879 
9.781 

10.912 
4.136 
1.974 


12.5 

50 

50 
100 
100 
100 
100 
100 


1.32 


2 


Renewals of rails 


.72 


3 


Renewals of ties 


1.55 


12 

21 


Repairs of locomotives 

Enginemen 


5.88 
9.78 


22-25 


Engine supplies 


10.91 


28 . 


Switchmen, etc 


4.14 


29 


Telegraph, 


1.97 










36.27 



451. Numerical comparison of pusher and through grades. 
In § 445 the computation was made of the desirability of re- 
ducing a 1.6% ruling grade to a 1.2% grade. Suppose it is 
found that by keeping the 1.6% grades as pusher grades having 
a total length of 20 miles on a 100 mile division, the other grades 
may be reduced to a grade not exceeding 0.713% (the correspond- 
ing through grade) for an expenditure of $200000. Will it 
pay? The saving by cutting down trains from 8 to 4, computed 
as before, would be (see §445) 4 X $0.5333X100X365 =$77862. 
But this saving is only accomplished by the employment of 
pushers making four round trips over 20 miles of pusher grades 
at a cost of 4x20x8260 =$20800. 

The net annual saving is therefore $57062, which when 
capitalized at 5% =$1,141,240. 

The above estimate probably has this defect. The total 
daily pusher-engine mileage is but 2X4X20 = 160, scarcely 
work enough for two pushers. Unless the pusher grades were 
bunched into two groups of about 10 miles each, two pusher 
engines could not do the work. If the number of trains was 
much larger, then the above method of calculation would be 
more exact even though the 20 miles of pusher grade was divided 
among four or five different grades. Therefore with the above 
data the annual cost of the pusher service would probably be 
much more — perhaps twice as much — and the annual saving 
about $36000, which would justify an expenditure of $720000. 
But even this would very amply justify the assumed expenditure 
of $200000 which would accomplish this result. 

The above computation is but an illustration ot the general 



§ 452. GRADE. 489 

truth which has been previously stated. In spite of the un- 
certainties and the variations of many items in the above esti- 
mates it will generally be possible to make a computation which 
Avill show unquestionably, as in the above instance, what is 
the best and the most economical method of procedure. When 
the capitalized valuations of both m.ethods are so nearly equal 
that a proper choice is more difficult, the question will frequently 
be determined by the relative ease of raising additional capital. 

BALANCE OF GRADES FOR UNEQUAL TRAFFIC. 

452. Nature of the subject. It sometimes happens, as when 
a road runs into a mountainous country for the purpose of 
hauling therefrom the natural products of lumber or minerals, 
that the heavy grades are all in one direction — that the whole 
line consists of a more or less unbroken climb having perhaps 
a few comparatively level stretches, but no do^Mi grade (except 
possibly a slight sag) in the direction of the general up grade. 
With such lines this present topic has no concern. But the 
majority of railroads have termini at nearly the same level 
(500 feet in 500 miles has no practical effect on grade) and 
consist of up and down grades in nearly equal amounts and 
rates. The general rate of ruling grade is determined by the 
character of the country and the character and financial backing 
of the road to be built. It is always possible to reduce the grade 
at some point by ^'development" or in general by the expen- 
diture of more money. It has been tacitly assumed in the 
previous discussions that when the ruling grade has been de- 
termined aU grades in either direction are cut down to that 
limit. If the traflic in both directions were the same this would 
be the proper policy and sometimes is so. But it has developed, 
especially on the great east and west trunk lines, that the weight 
of the east bound freight traffic is enormously greater than that 
of the westbound — that westbound trains consist ver}^ largely of 
*' empties'' and that an engine which could haul twenty loaded 
cars up a given grade in eastbound traflic could haul the same 
cars empty up a much higher grade when running west. As 
an illustration of the large disproportion which may exist, the 
eastbound ton-mileage on the P. R. R. between the years 1851 
and 1885 was 3.7 times the westbound ton-mileage. Between 
the years 1876 and 1880 the ratio rose to more than 4.5 to 1. 



490 RAILROAD CONSTRUCTION. § 453. 

On such a basis it is as important and necessary to obtain, say, 
a 0.6% ruling grade against the eastbound trafhc as to have, 
say, a 1.0% grade against the westbound trafiic. This is the 
basis of the foilov;ing discussion. It now remains to estimate 
the probable ratio of the traffic in the two directions and from 
that to determine the proper ^'balance'' of the opposite ruling 
grades. 

453' Computation of the theoretical balance. Assume first, 
for simplicity, that the exact business in either direction i£; 
accurately known. A little thought will show the truth of the 
following statements. 

1. The locomotive and passenger-car traffic in both directions 
is equal. 

2. Except as a road may carry emigrants, the passenger 
traffic in both directions is equal. Of course there are innumer- 
able indiA'idual instances in which the return trip is made by 
another route, but it is seldom if ever that there is any marked 
tendency to uniformity in this. Considering that a car load 
of, say, 50 passengers at 150 pounds apiece weigh but 7500 
pounds, which is J of the 45000 pounds which the car may 
weigh, even a considerable variation in the number of passengers 
wiU not appreciably affect the haulirg of cars on grades. On 
parlor-cars and sleepers the ratio of live load to dead load (say 
20 passengers, 3000 pounds, and the car, 75000 pounds) is 
even more insignificant. The effect of passenger traffic on 
balance of grades may therefore be disregarded. 

3. Empty cars have a greater resistance jper ton than loaded 
cars. Therefore in computing the hauling capacity of a loco- 
motive hauling so many tons of ^' empties," a larger figure must 
be UGcd for the ordinary tractive resistances — say two pounds 
per ton greater. 

4„ Owing to greater or less imperfections of management a 
small percentage of cars will run empty or but partly full in 
the direction of greatest traffic. 

5. Freight having great bulk and weight (such as grain, 
lumber, coal, etc.) is run from the rural districts toward the 
cities and manufacturing districts. 

6. The return traffic — manufactured products — although worth 
as much or more, do not w^eigh as much. 

As a siniDle numerical illustration assume that the weight 
C'f the cars is 40% and the live load 60% of the total load when 



§ 454. GRADE. 491 

th3 cars are ^^fiill" — although not loaded to their absolute 
liinit of capacity. Assume that the relative weight of live 
load to be hauled in the other direction is but J. Then the 
gross train load (exclusive of the locomotive) is 40 + (^ X 60) =60% 
of the load in the other direction. Assume that the grade 
against the heaviest traffic is 0.9%. An 80-ton engine with 48 
tons on the drivers, ^ adhesion, normal tractive resistance 
6 pounds per ton, will haul a train of 920 tons up that grade. 
Of this load the cars are assumed to weigh 40%, or 368 tons, 
and the live load 552 tons. On the return trip the weight of 
the cars with their load is but 920X60% =552 tons, or with the 
engine 632 tons. This could be hauled up a 1.60% grade, as- 
suming that the resistance was the same per ton. But f of the 
return cars must be figured as empty; they make an added 
resistance of 2 pounds per ton; these cars weigh f of 368 tons, 
or 245.33 tons. The balance of the train weighs 632-245.33 = 
386.67 tons. Then we have 

245.33X8=1962.67 
386.67X6=2320. 



4282.67 pounds, 

which is the tractive force required for rolling resistances, etc. 
Subtracting this from the total adhesion, 24000 pounds, we have 
left 19717.33 pounds available for grade, or 31.2 pounds per 
ton, which corresponds to a 1.56% grade, which is the proper 
balance of grade under the above conditions. 

454. Computation of relative traffic. Some of the principal 
elements have already been referred to, but in addition the 
foUomng facts should be considered. 

(a) The greatest disparity in traffic occurs through the hand- 
hng of large amounts of coal, lumber, iron ore, grain, etc. On 
roads which handle but little of these articles or on which for 
local reasons coal is hauled one way and large shipments of 
grain the other way the disparity will be less and will perhaps be 
insignificant. 

(b) A marked change in the development of the country may, 
and often does, cause a marked difference in the disparity of 
traffic. The heaviest traffic (in mere weight) is always toward 
manufacturing regions and away from agricultural regions. But 
when a region, from being purely agricultural or mineral, be^ 



492 RAILROAD CONSTRUCTION. § 454. 

comes largely manufacturing, or when a manufacturing region 
develops an industry which wiU cause a growth of heavy freight 
traffic from it, a marked change in the relative freight movement 
will be the result. 

(c) Very great fluctuations in the relative traffic may be 
expected for prolonged intervals. 

(d) An estimate of the relative traffic may be formed b}^ 
the same general method used in computing the total traffic 
of the road (see § 373, Chap. XIX) or by noting the relative 
traffic on existing roads which may be assumed to have practically 
the same traffic as the proposed road will obtain. 



i\ 



CHAPTER XXIV. 

THE IMPROVEMENT OF OLD LINES. * 

455. Classification of improvements. The improvements here 
considered are only those of alignment — horizontal and vertical. 
vStrictly there is no definite limit, either in kind or magnitude, 
to the improvements which may be made. But since a railroad 
cannot ordinarily obtain money, even for improvements^ to 
an amount greater than some small proportion of the pre- 
viously invested capital, it becomes doubly necessary to expend 
such money to the greatest possible advantage. It has been 
pre\dously shown that securing additional business and increas- 
ing the train load are the two most important factors in decreas- 
ing dividends. After these, and of far less importance, come 
reductions of curvature, reductions of distance (frequently of 
doubtful policy, see Chap. XXI, §414), and elimination of sags 
and humps. These various improvements will be briefly dis- 
cussed. 

(a) Securing additional business. It is not often possible 
by any small modification of alignment to materially increase 
the business of a road. The cases which do occur are usually 
those in which a gross error of judgment was committed during 
the original construction. For instance, in the early history 
of railroad construction many roads were largely aided by the 
towns through which the road passed, part of the money neces- 
sary for construction being raised by the sale of bonds, wliich 
were assumed or guaranteed and subsequently paid by the 
toTVTis. Such aid was often demanded and exacted by the 
promoters. Instances are not unknown where a failure to 
come to an agreement has caused the promoters to deliberately 
pass by the to\\Ti at a distance of some miles, to the mutual 
disadvantage of the road and the town. If the town subsequent- 
ly grew in spite of this disadvantage, the annual loss of business 
might readily amount to more than the original sum in dispute. 

493 



494 RAILROAD CONSTRUCTION. § 456. 

Such an instance would be a legitimate opportunity for study 
of the advisability of a re-location. 

As another instance (the original location being justifiable) 
a railroad might have been located along the bank of a consider- 
able river too wide to be crossed except at considerable expense. 
When originally constructed the enterprise would not justify 
the two extra bridges needed to reach the town. A growth in 
prosperity and in the business obtainable might subsequently 
make such extra expense a profitable investment. 

(b) Increasing the train load. On account of its importance 
this will be separately considered in § 458 et seq. 

(c) Reductions in curvature and distance and the elimination 
of sags and humps. The financial value of these improvements 
has already been discussed in Chapters XXI, XXII, and XXIII. 
Such improvements are constantly being made by all progressive 
roads. The need for such changes occurs in some cases because 
the original location was very faulty, the revised location being 
no more expensive than the original, and in other cases because 
the original location was the best that was then financially 
possible and because the present expanded business will justify 
a change. 

(d) Changing the location of stations or of passing sidings. 
The station may sometimes be re-located so as to bring it nearer 
to the business center and thus increase the business done. 
But the principal reasons for re-locating stations or passing 
sidings is that starting trains may have an easier grade on which 
to overcome the additional resistances of starting. Such changes 
will be discussed in detail in § 460, 

456. Advantages of re-locations. There are certain undoubted 
advantages possessed by the engineer who is endeavoring to 
improve an old line. 

(a) The gross traffic to be handled is definitely kno^Ti. 

(b) The actual cost per train-mile for that road (which may 
differ very greatly from the average) is also known, and therefore 
the value of the proposed improvement can be more accurately 
determined. 

(c) The actual performance of such locomotives as are used 
on the road may be studied at leisure and more reliable data 
may be obtained for the computations. 

457. Disadvantages of re-locations. The disadvantages are 
generally more apparent and frequently appear practically 



^ 457. IMPROVEMENT OF OLD LINES. 495 

I 

linsuperable — more so than they prove to be on closer inspection, 
a) It frequently means the abandonment of a greater or less 
ength of old line and the construction of new line. At first 
thought it might seem as if a change of line such as would permit 
lan increase of train-load of 50 or perhaps 100% could never 
he obtained, or at least that it could not be done except at an 
impracticable expense. On the contrary a change of 10% 
|iof the old line is frequently all that is necessary to reduce the 
'grades so that the train-loads hauled by one engine ma}^ be 
^1 nearly if not quite doubled. And when it is considered that 
, the cost of a road to sub-grade is generally not more than one- 
t third of the total cost of construction and equipment per mile, 
j'it becomes plain that an expenditure of but a. small percentage 
of the original outlay, expended w^here it will do the most good, 
'iwill often suffice to increase enormousty the earning capacity. 
I (h) One of the most difficult matters is to convince the finan- 
Icial backers of the road that the proposed improvement will 
' be justifiable. The cause is simple. The disadvantages of the 
original construction lie in the large increase of certain items 
of expense which are necessary to handle a given traffic. And 
t yet the fact that the expenditures are larger than they need 
, be are only apparent to the expert, and the fact that a saving 
may be made is considered to be largely a matter of opinion 
until it is demonstrated by actual trial. On the other hand 
I the cost of the proposed changes is definite, and the very fact 
that the road has been uneconomically worked and is in a poor 
fi^nancial condition makes it difficult to obtain money for im- 
provements. 

(c) The legal right to abandon a section of operated line 
and thus reduce the value of some adjoining propert}^ has 
sometimes been successfully attacked. A common instance 
would be that of a factory which w^as located adjoining the right 
of way for convenience of transportation facilities. The abandon- 
ment of that section of the right of way would probabl}' be fatal 
to the successful operation of the factory. The objection may 
be largely eliminated by the maintenance of the eld right of 
way as a long siding (although the business of the factory might 
not be worth it) , but it is not alwa3^s so easy of solution, and 
this phase of the question must always be considered. 



496 RAILROAD CONSTRUCTION. § 458. 



REDUCTION OF VIRTUAL GRADE. 

458. Obtaining data for computations. As developed in the 
last chapter (§§ 432-434) the real object to be attained is the 
reduction of the virtual grade. The method of comparing grades 
under various assumed conditions was there discussed. When 
the road is still '^on paper'' some such method is all that is 
possible; but when the road is in actual operation the virtual 
grade of the road at A^arious critical points, w^th the rolling 
stock actually in use. may be determined by a simple test and 
the effect of a proposed change may be reliably computed. 
Bearing in mind the general principle that the \drtual grade 
line is the locus of points determined by adding to the actual 
grade profile ordinates equal to the velocity head of the train, 
it only becomes necessary to measure the velocity at various 
points. Since the velocity is not usually uniform, its precise 
determination at an^^ instant is almost impossible, but it will 
generally be found to be sufficiently precise to assume the velocity 
to be uniform for a short distance, and then observe the time 
required to pass that short space. Suppose that an ordinary 
watch is used and the time taken to the nearest second. At 
30 railes per hour, the velocity is 44 feet per second. To obtain 
the time to within 1%, the time would need to be 100 seconds 
and the space 4400 feet. But with variable velocity there 
would be too great error in assuming the velocity as uniform 
for 4400 feet or for the time of 100 seconds. Using a stop- 
watch registering fifths of a second, a 1% accuracy would 
require but 20 seconds and a space of 880 feet, at 30 miles per 
hour. Wellington suggests that the space be made 293 feet 
4 inches, or -^-^ of a mile; then the speed in miles per hour 
equals 200 -^ s, in which s is the time in seconds required to 
traverse the 293' 4''. For instance, suppose the time required 
to pass the interval is 12.5 seconds. jV mile in 12.5 seconds = 
one mile in 225 seconds, or 16 miles per hour. But likewise 
200 -^ 12.5 =16, the required velocity. The following features 
should be noted when obtaining data for the computations: 

(a) All critical grades on the road should be located and 
their profiles obtained — by a survey if necessary. 

(6) At the bottom and top of all long grades (and perhaps at 
intermediate points if the grades are very long) spaces of known 



§ 459. IMPROVEMENT OF OLD LINES. 497 

length (preferably 293 J feet) should be measured off and marked 
by flags, painted boards, or any other ser\dceable targets. 

(c) Provided with a stop-watch marking fifths of seconds 
the observer should ride on the trains affected by these grades 
and note the exact interval of time required to pass these spaces. 
If the 'space is 293 J feet, the velocity in miles per hour =200 -f- 
interval in seconds. In general, 

^. distance in feet X 3600 



time in seconds X 5280* 



(d) Since these critical grades are those which require the 
greatest tax on the power of the locomotive, the conditions 
under which the locomotive is working must be knowTi — i.e., 
the steam pressure, point of cut-off, and position of the throttle. 
Economy of coal consumption as well as efficient working at 
high speeds requires that steam be used expansively (using an 
earl}- cut-off), and even that the throttle be partly closed; but 
when an engine is slowh^ chmbing up a maximum grade ^Yith a 
full load it is not exerting its maximum tractive power unless 
it has its maximum steam pressure, wide-open throttle, and is 
cutting off nearly at full stroke. These data must therefore 
be obtained so as to know whether the engine is dcA^eloping 
at a critical place all the tractive force of which it is capable. 
The condition of the track (wet and slippery or dr}^) and the 
approximate direction and force of the wind should be noted 
with sufficient accuracy to judge whether the test has been made 
under ordinary conditions rather than under conditions which 
are exceptionally favorable or unfavorable. 

(e) The train-loading should be obtained as closely as possible. 
Of course the dead weight of the cars is easii}^ found, and the 
records of the freight department will usually giA^e the live 
load with aU sufficient accuracy. 

459. Use of the data obtained. A ver}^ brief inspection 
of the results, freed from refined calculations or uncertainties, 
will demonstrate the following truths: 

(a) If, on a uniform grade, the velocity increases, it shows 
that, under those conditions of engine working, the load is less 
than the engine can handle on that grade 

(h) If the velocity decreases, it shows that the load is greater 
than the engine can handle on an indefinite length of such 



498 RAILROAD CONSTRUCTION. § 459. 

grade. It shows that such a grade is being operated by momen- 
tum. From the rate of decrease of velocity the maximum 
practicable length of such a grade (starting with a given velocity) 
may be easily computed. 

(c) By combining results under different conditions of grade 
but with practically the same engine working, the tractiA^e 
power of the engine may be determined (according to the prin- 
ciples previously demonstrated) for any grade and velocity. 
For example: On an examination of the profile of a division 
of a road the maximum grade w^as found to be 1.62% (85.54 
feet per mile). At the bottom and near the top of this grade 
two lengths of 293' 4'' are laid off. The distance between the 
centers of these lengths is 6000 feet. A freight train moving 
up the grade is timed at 9f seconds on the lower stretch and 7f 

seconds on the upper. These times correspond to — — and ,^— ^ 

or 21.3 and 26.3 miles per hour respectively. It is at once 

observed that the velocit}^ has increased and that the engine 

could draw even a heavier load up such a grade for an indefinite 

distance. How^ much heavier might the load be? 

For simplicity we will assume that the conditions were 

normal, neither exceptionally favorable nor unfavorable, and 

that the engine was worked to its maximum capacity. The 

engine is a ^'consolidation" weighing 128700 pounds, with 

112600 pounds on the drivers. The train-load behind the 

engine consists of ten loaded cars weighing 465 tons and eleven 

empties w^eighing 183 tons, thus making a total train-weight of 

712 tons. Applying Eq. 140, we find that the additional force 

which the engine has actually exerted per ton in increasing the 

velocity from 21.3 to 26.3 miles per hour in a distance of 6000 

feet is 

70 224 
P=-^g^(26.32-21.32) =2.78 pounds per ton 

The grade resistance on a 1.62% grade is 32.4 pounds per 
ton. With an average train resistance of seven pounds per ton 
the total necessary pull for uniform velocity would be 39.4 
pounds. But the engine is actually exerting an additional pull 
of 2.78 pounds per ton. Evidently its total load might be 
increased proportionately, i.e., the total train-load might equal 

712X^-^|±|^=762tons. 



§ 460. IMPROVEMENT OF OLD LINES. 499 

This shows that 50 tons additional might have been loaded 
on, say, three more empties or one loaded car and one empty^ 
An overload of a few tons would easily be made up by a very 
slight reduction in the velocity. 

The above calculation should of course be considered simply 
as a ''single observation.'' The performance of the same engine 
on the same grade (as well as on many other grades) on succeed- 
ing days should also be noted. It may readily happen that 
variations in the condition of the track or of the handling of the 
engine may make considerable variation in the results of the 
several calculations, but when the work is properly done it is 
alwa}'s possible to draw definite and very positive deductions. 

460. Reducing the starting grade at stations. The resistance 
to starting a train is augmented from two causes: (a) the trac- 
tive resistances are usually about 20 pounds pei' ton instead 
of, say, 6 pounds, and (b) the inertia resistance must be overcome. 
The inertia resistance of a freight train (see § 347) w^hich is 
expected to attain a velocity of 15 miles per hour in a distance 
of 1000 feet is (see Eq. 140) 

70 224 
P = (15^ — 0) =15.8 pounds per ton, which is the equiva- 

lent of a 0.79% grade. Adding this to a grade which nearly or 
quite equals the ruling grade, it virtvally creates a new and 
higher ruling grade. Of course that additional force can be 
greatly reduced at the expense of slower acceleration, but even 
this cannot be done indefinitely, and an acceleration to only 
15 miles per hour in 1000 feet is as slow as should be allowed 
for. With perhaps 14 pounds per ton additional tractive 
resistance, we have about 30 pounds per ton additional — equiva- 



FiG. 217. 



lent to a 1.5% grade. Instances are^ known where it has proven 
wise to create a hump (in what was otherwise a uniform grade) 



500 RAILROAD CONSTRUCTION. § 460. 

at a station. The effect of this on high-speed passenger trains 
moving up the grade would be merely to reduce their speed 
very slightly. No harm is done to trains moving down the 
grade. Freight trains moving up the grade and intending to 
stop at the station will merely have their velocity reduced as 
they approach the station and will actually save part of the 
wear and tear otherwise resulting from applying brakes. When 
the trains start they are assisted by the short down grade, 
jtist where they need assistance most. Even if the grade CD 
is still an up grade, the pull required at starting is less than that 
required on the uniform grade b}' an amount equal to 20 times 
the difference of the grade in per cent. o 

a 



APPENDIX. 

THE ADJUSTMENTS OF mSTKUMENTS. 

The accuracy of instrumental work may be vitiated by any 

one of a large number of inaccuracies in the geometrical relations 

of the parts of the instruments. Some of these relations are so 

apt to b^ rltered by ordinary usage of the instrument that the 

makers have provided adjusting-screws so that the inaccuracies 

J may be readily corrected. There are other possible defects, 

r| which, however, will seldom be found to exist, provided the 

I! instrument was properly made and has never been subjected to 

' treatment sufficiently rough to distort it. Such defects, when 

, found, can only be corrected by a competent instrument-maker 

il or repairer. 

A WARNING is necessary to those who would test the accuracy 

,, cf instruments, and especially to those whose experience in such 

' work is small. Lack of skill in handling an instrument Tvdll 

often indicate an apparent error of adjustment when the real 

(i error is very different or perhaps non-existent. It is always a 

' safe plan when testing an adjustment to note the amount of the 

apparent error; then, beginning anew, make another independent 

Ij determination of the amount of the error. When two or more 

I perfectly independent determinations of such an error are made 

it will generally be found that they differ by an appreciable 

amount. The differences may be due in variable measure to 

careless inaccurate manipulation and to instrumental defects 

which are wholly independent of the particular test being made. 

Such careful determinations of the amounts of the errors are 

generally advisable in view of the next paragraph. 

Do NOT DISTURB THE ADJUSTING-SCREWS ANY MORE THAN 

NECESSARY. Although metals are apparently rigid, they are 
really elastic and yielding. If some parts of a complicated 
mechanism, which is held together largely by friction, are sub- 
jected to greater internal stresses than other parts of the mech- 

501 



502 RAILROAD CONSTRUCTION. 

anism, the jarring resulting from handling will frequently cause 
a slight readjustment in the parts which will tend to more nearly 
equalize the internal stresses. Such action frequently occurs 
with the adjusting mechanism of instruments. One screw may 
be strained more than others. The friction of parts may pre- 
vent the opposing screw from immediately taking up an equal 
stress. Perhaps the adjustment appears perfect under these 
conditions Jarring diminishes the friction between the parts, 
and the unequal stresses tend to equalize. A motion takes place 
which, although microscopically minute, is sufficient to indicate 
an error of adjustment. A readjustment made by unskillful 
hands ma,y not make the final adjustment any more perfect. 
The frequent shifting of adjusting-screws wears them badly, 
and when the screws are worn it is still more difficult to keep 
them from moving enough to vitiate the adjustments. It is 
therefore preferable in many cases to refrain from disturbing the 
adjusting-screws, especially as the accuracy of the work done is 
not necessarily affected by errors of adjustment, as may be 
illustrated: 

(a) Certain operations are absolutely unaffected by certain 
errors of adjustment. 

(b) Certain operations are so slightly affected by certain small 
errors of adjustment that their effect may properly be neglected. 

(c) Certain errors of adjustment may be readily allowed for 
and neutralized so that no error result <=! from the use of the 
unadjusted instrument. Illustrations of all these cases will be 
given under their proper heads. 

ADJUSTMENTS OF THE TRANSIT. .|& 

1. To hare the plafe-bubhles in the center of the tubes when the 
axis is vertical. Clamp the upper plate and, with the lower 
clamp loose, swing the instrument so that the plate-bubbles are 
parallel to the lines of opposite leveling-screws. Level up until 
both bubbles are central. Swing the instrument 180°. If the 
bubbles again settle at the center, the adjustment is perfect. If 
either bubble does not settle in the center, move the leveling- 
screws until the bubble is half-way back to the center. Then, 
before touching the adjusting-screws, note carefully the position 
of the bubbles and observe whether the bubbles always settle at 
the same place in the tube, no matter to what position the in- 



APPENDIX, 503 

striiment may be rotated. When the instrument is so leveled, 
the axis is truly vertical and the discrepancies between this 
constant position of the bubbles and the centers of the tubes 
measure the errors of adjustment. By means of the adjusting- 
screws bring each bubble to the center of the tube. If this is 
done so skillfully that the true level of the instrument is not 
disturbed, the bubbles should settle in the center for all positions 
of the instniment. Under unskillful hands, two or more such 
trials may be necessary. 

When the plates are not horizontal, the measured angle is greater than 
the true horizontal angle by the difference between the measured angle 
and its projection on a horizontal plane. When this angle of inclination 
is small, the difference is insignificant. Therefore when the plate-bubbles 
are very nearly in adjustment, the error of measurement of horizontal 
angles may be far within the lowest unit of measurement used. A small 
error of adjustment of the plate-bubble perpendicular to the telescope will 
affect the horizontal angles by only a small proportion of the error, vrhich 
will be perhaps imperceptible. Vertical angles will be affected by the 
same insignificant amount. A small error of adjustment of the plate- 
bubble parallel to the telescope will affect horizontal angles very slightly, 
but w^ill affect vertical angles by the full amount of the error. 

All error due to unadjusted plate-bubbles may be avoided by noting in 
what positions in the tubes the bubbles will remain fixed for all positions 
of azimuth and then keeping the bubbles adjusted to these positions, for 
the axis is then truly vertical. It will often save time to work in this way 
temporarily rather than to stop to make the adjustments. This should 
especially be done when accurate vertical angles are required. 

When the bubbles are truly adjusted, they should remain stationary 
regardless of whether the telescope is revolved with the upper plate loose 
and the lower plate clamped or whether the whole instrument is revolved, 
the plates being clamped together. If there is any appreciable difference, 
it shows that the two vertical axes or "centers" of the plates are not con- 
centric. This may be due to cheap and faulty construction or to the exces- 
sive wear that may be sometimes observed in an old instrument originally 
well made. In either case it can only be corrected by a maker. 

2. To make ike revolving axis of the telescope perpendicular to 
the vertical axis of the instrument. This is best tested by using 
a long pluml.)-line, so placed that the telescope must be pointed 
upward at an angle of about 45° to sight at the top of the plumb- 
line and dowmw^ard about the same amount, if possible, to 
sight at the lower end. The vertical axis of the transit must 
be made truly vertical. Sight at the upper part of the line; 
clamping the horizontal plates. Swing the telescope down 
and see if the cross-wire again bisects the cord. If so, the 
adjustment is probably perfect (a conceivable exception will be 



1 



504 RAILROAD CONSTRUCTION. 

noted later) ; if not, raise or lower one end of the axis by means 
of the adjusting-screws, placed at the top of one of the standards, 
until the cross-mre Y>dll bisect the cord both at top and bottom. 
The plumb-bob may be steadied, if necessar}^, by hanging it 
in a pail of water. As maiBy telescopes cannot be focused 
on an object nearer than G or 8 feet from the telescope, this 
method requires a long plumb-line swung from a high point, 
which may be inconvernent. 

Another method is to set up the instrument about 10 feet 
from a high wall. After levehng, sight at some convenient 
mark high up on the wall. Swing the telescope down and make 
a mark (when working alone some convenient natural mark may 
generally be found) low down on the wall. Plunge the telescope 
and revolve the instrument about its vertical axis and again sight 
at the upper mark. Swing down to the lower mark. If the 
wire again bisects it, the adjustment is perfect. If not, fix a 
point half-way between the two positions of the lower mark. 
The plane of this point, the upper point, and the center of the 
instrument is trul}^ vertical. Adjust the axis to these upper and 
lower points as when using the plumb-line. 

3. To make the line of collimation perpendicular to the revolving 
axis of the telescope. With the instrument level and the telescope 
nearly horizontal point at some well-defined point at a distance 
of 200 feet or more. Plunge the telescope and establish a point 
in the opposite direction. Turn the whole instrument about the 
vertical axis until it again points at the first mark. Again 
plunge to ^'direct position'' (i.e., vnih the level-tube under 
the telescope). If the vertical cross- wire again points at the 
second mark, the adjustment is perfect. If not, the error is 
one-fourth of the distance between the two positions of the 
second mark. Loosen the capstan screw on one side of the 
telescope and tighten it on the other side until the vertical 
wire is set at the one-fourth mark. Turn the whole instrument 
by means of the tangent screw until the vertical wire is midway 
between the two positions of the second mark. Plunge the 
telescope. If the adjusting has been skillfully done, the cross- 
wire should come exactly to the first mark. As an ''erecting 
eyepiece" reinverts an image already inverted, the ring carr3qng 
the cross-wires must be moved in the saine direction as the; 
apparent error in order to correct that error. 



APPENDIX. 505 

The necessity for the third adjustment lies principally in the practice 
of producing a line by plunging the telescope, but when this is required to 
be done with great accuracy it is always better to obtain the forward point 
by reversion (as described above for making the test) and take the mean 
of the two forward points. Horizontal and vertical angles are practically 
unaffected by small errors of this adjustment, unless, in the case of hori- 
zontal angles, the vertical angles to the points observed are very different. 

Unnecessary motion of the adjusting-screws may sometimes be avoided 
by carefully establishing the forward point on line bj^ repeated reversions 
of the instrument, and thus determining by repeated trials the exact amount 
of the error. Differences in the amount of error determined would be 
evidence of inaccuracy in manipulating the instrument, and would show 
that an adjustment based on the first trial would probably prove unsatis- 
factory. 

The 2d and 3d adjustments are mutually dependent. If either adjust- 
ment is badly out, the other adjustment cannot be made except as follows: 

(a) The second adjustment can be made regardless of the third when 
the lines to the high point and the low point make equal angles with the 
horizontal. 

{b) The third adjustment can be made regardless of the second when 
the front and rear points are on a level with the instrument. 

When both of these requirements are nearly fulfilled, and especially 
when the error of either adjustment is small, no trouble will be found in 
perfecting either adjustment on account of a small error in the other ad- 
justment. 

If the test for the second adjustment is made by means of the plumb- 
line and the vertical cross-wire intersects the line at all points as the tele- 
scope is raised or lowered, it not only demonstrates at once the accuracy 
of that adjustment, but also shows that the third adjustment is either 
perfect or has so small an error that it does not affect the second. 

4. To have the huhhle of the telescope-level in the center of the 
tube when the line of collimation is horizontal. The line of coUi- 
mation should coincide with the optical axis of the telescope. 
If the object-glass and eyepiece haA^e been properly centered, 
the previous adjustment will have brought the vertical cross- 
wire to the center of the field of view. The horizontal cross- 
wire should also be brought to the center of the field of \4ew, 
and the bubble should be adjusted to it. 

a. Peg method. Set up the transit at one end of a nearly 
level stretch of about 300 feet. Clamp the telescope with its 
bubble in the center. Drive a stake vertically under the eye- 
]3iece of the transit, and another about 300 feet away. Observe 
the height of the center of the eyepiece (the telescope being 
level) above the stake (calling it a) ; obser\'e the reading of the 
rod when held on the other stake (calling it b) ; take the instru- 
ment to the other stake and set it up so that the eyepiece is 



506 RAILROAD CONSTRUCTION. 

vertically over the stake, observing the height, c ; take a reading 
on the first stake, calling it d. If this adjustment is perfect, 
then 

a— c?=6 — c, 
or (a-'d)-{h—c)=Q. 

Call {a-d)-{h-c)^2m. 
When m is positive, the line points downward ; 
" m '* negative, " '^ '' upward. 

To adjust: if the line points wp, sight the horizontal cross- 
wire (by moving the vertical tangent screw) at a point which is 
m lower, then adjust the bubble so that it is in the center. 

By taking several independent values for a, h, c, and dj a mean value 
for m is obtained, which is more reliable and which may save much un- 
necessary working of the adjusting-screws. 

h. Using an auxiliary level. When a carefully adjusted level 
is at hand, this adjustment may sometimes be more easily 
made by setting up the transit and level, so that their lines of 
coUimation are as nearly as possible at the same height If a 
point may be found which is half a mile or more away and 
which is on the horizontal cross-wire of the level, the horizontal 
cross-wire of the transit may be pointed directly at it, and the 
bubble adjusted accordingly. Any slight difference in the 
heights of the lines of coUimation of the transit and level (say 
\'^) may almost be disregarded at a distance of J mile or more, 
or, if the difTerence of level would have an appreciable effect, 
even this may be practically eliminated by making an estimated 
allowance when sighting at the distant point. Or, if a distant 
point is not available, a level-rod Tidth target may be used at a 
distance of (say) 300 feet, making allowance for the carefully 
determined difference of elevation of the two lines of coUimation. 

5. Zero of vertical circle. When the line of coUimation is truly 
horizontal and the vertical axis is truly vertical, the reading 
of the vertical circle should be 0°. If the arc is adjustable, 
it should be brought to 0°. If it is not adjustable, the index 
error should be observed, so that it may be applied to all readings 
of vertical angles. 

ADJUSTMENTS OF THE WYE LEVEL. 

1. To make the line of coUimation coincide with the center of 
the rings. Point the intersection of the cross-wires at some 



APPENDIX. ^"' 

well-defined point which is at a considerable distance The in- 
strument need not be level, which allows much greater liberty 
in choosing a convenient point. The vertical axis should be 
clamped, and the clips over the wyes should be loosened and 
raised. Rotate the telescope in the wyes. The intersection of 
the cross-wires should be continually on the point If it is not 
t requires adjustment. Rotate the telescope 180 and adjus 
or^-half of the error by means of the capstan-headed screws that 
move the cross-wire ring. It should be remembered tiiat, with 
an erecting telescope, on account of the inversion of the image, 
the ring should be moved in the direction of the apparent error. 
./Adjust the other half of the error with the leveling-screws 
Then rotate the telescope 90° from its usual position, .sight 
accurately at the point, and then rotate 180° from that posiUon 
and adjust any error as before. It may ''.^q^^'-^ ^f ^'■'^\.'"/^'' 
but it is necessary to adjust the ring until the intersection o 
the cross-^vires will remain on the point for any position of 
rotation. 

If such a test is made on a very distant point and again on a point only 
10 orltfeetfom the instrument, the adjustment may be found correct 
for one pott and incorrect for the other. Thi« indicates that the object- 
:rL"ts%Lproperiy centered, .s^^^^^^^^^^^^^ 

Tutr Smrshortntrma^e on aU which is at about the 
mean distance for usual practice— say 150 feet. . , ,„. ;. ;„Hi- 

Tf the whole ima^e appears to shift as the telescope ,s rotated it indi 
,te, thalthe eyepiece is improperly adjusted. This defect is likewise 

to be eccentric with the field of view. 

2 To rmke the axis of the levd-tvhe parallel to the line of colli. 
\naHo^v. Raise the clips as far as possible. Swing the levd 
so that it is paraUel to a pair of opposite levehng-screws and 
cLmp it. Bring the bubble to the middle of the tube by means 
fe oFt'e levehng-screws. Take the telescope out of the wyes anrt 
^ r place it end for end, using extreme care that the wyes are not 
iarred by the action. If the bubble does not come o the center 
correct one-half of the error by the vertical adjustmg-screws at 
one end of the bubble. Correct the other half by the levehng- 
screws. Test the work by again changing the telescope end for 

"""carVlhouW be taken while making this adjustment to see 



h 



508 RAILROAD CONSTRUCTION, 

that the level-tube is vertically under the telescope. With the 
bubble in the center of the tube, rotate the telescope in the wyes 
for a considerable angle each side of the vertical. If the first 
half of the adjustment has been made and the bubble moves, it 
shows that the axis of the wyes and the axis of the level-tube 
are not in the same vertical plane although both have been made 
horizontal. By moving one end of the level-tube sidewise by 
means of the horizontal screws at one end of the tube, the two 
axes may be brought into the same plane. As this adjustment 
is liable to disturb the other, both should be alternately tested 
until both requirements are complied \^dth. 

By these methods the axis of the bubble is made parallel to 
the axis of the wj^es; and as this has been made parallel to the 
lines of collimation by means of the previous adjustment, the 
axis of the bubble is therefore parallel to the line of collimation. 

3, To make the line of collimation perpendicular to the vertical 
axis. Level up so that the instrument is approximately level 
over both sets of leveling-screws. Then, after leveling carefully 
over one pair of screws, revolve the telescope 180° If it is not 
level, adjust half of the error by means of the capstan-headed 
screw under one of the wyes, and the other half by the leveling- 
screws. Reverse again as a test. . 

When the first two adjustments have been accurately made, good level- 
ing may always be done by bringing the bubble to the center by means of 
the leveling-screws, at every sight if necessary, even if the third adjust- 
ment is not made. Of course this third adjustment should be made as a 
matter of convenience, so that the line of collimation may be always level 
no matter in what direction it may be pointed, but it is not necessary to 
stop work to make this adjustment every time it is found to be defective. 

ADJUSTMENTS OF THE DUMPY LEVEL. 

1. To make the axis of the level-tube perpendicular to the vertical 
axis. Level up so that the instrument is approximately level 
over both sets of leveling-screws. Then, after leveling care- 
fully over one pair of screws, revolve the telescope 180°. If 
it is not level, adjust one-half of the error by means of the adjust- 
ing-screws at one end of the bubble, and the other half by 
means of the leveHng-screws. Reverse again as a test. 

2. To make the line of collimation perpendicular to the vertical 
axis. The method of adjustment is identical with that for 
the transit (No. 4, p. 505) except that the cross-wire must be 



APPENDIX. 509 

adjusted to agree with the level-bubble rather than vice versa, ag 
is the case with the corresponding adjustment of the transit; 
i.e., with the level-bubble in the center, raise or lower the hori- 
zontal cross-wire until it points at the mark known to be on 
a level with the center of the instrument. 

If the instrument has been well made and has not been dis- 
torted by rough usage, the cross-wires ^\ill intersect at the 
center of the field of view w^hen adjusted as described. If they 
do not, it indicates an error which ordinarily can only be cor- 
.rected by an instrument-maker. The error may be due to any 
one of several causes, which are 

(a) faulty centering of object-slide; 

(6) faulty centering of eyepiece; 

(c) distortion of instrument so that the geometric axis of 
the telescope is not perpendicular to the vertical axis. If the 
error is only just perceptible, it vrill not probably cause any 
error in the work. 



I 



EXPLANATORY NOTE ON THE USE OF THE TABLES. 

The logarithms here ghen arc ^^ five-place/' but the last 
figure sometimes has a special mark over it {e.g--, 6) Vvhich indi- 
cates that one-half a unit in the last place should be added. 
For example 



the value 
.69586 
.69586 



includes all values between 
.6958575000 + and .6958624999 
.6958625000 + and .6958674999 



The maximum error in any one value therefore does not 
exceed one-quarter of a fifth-place unit. 

When adding or subtracting such logarithms allow a half-unit 
for such a sign. For example 



.69586 


.69586 


.69586 


.10841 


.10841 


.10841 


.12947 


.12947 


.12947 



.93374 .93375 .93375 

All other logarithmic operations are performed as usual and 
are supposed to be understood by the student. 

511 









TABLE I.— RADII OF CURVES. 








Deg 


0° 




1° 


2° 


3° 


Deg 


Min 


Radius. 


Log 12 


Radius. 


Log 12 


Radius. 
2864-9 


Log 12 


Radius. 


LogU 


Min 





CO 


oo 


5729^6 


3-75813 


3-45711 


1910^1 


3.28105 





i 


343775 


5-53627 


5635^7 


.75095 


2841-3 


.45351 


1899^5 


•27864 


1 


2 


171887 


5-23524 


5544-8 


.74389 


2818-0 


.44993 


1889-1 


.27625 


2 


3 


114592 


5-05915 


5456-8 


•73694 


2795.1 


.44639 


1878-8 


-27387 


3 


4 


S5944 


4-93421 


5371^6 


•73010 


2772.5 


•44287 


1868-6 


-27151 


4 


5 


68755 


4-83730 


5288-9 


-72336 


2750-4 


•43939 


1858-5 


•26915 


5 


6 


57298 


4.75812 


5208-8 


3-71673 


2728^5 


3^43593 


1848-5 


3^26681 


6 


7 


49111 


•69117 


5131^0 


•71020 


2707.0 


•43249 


1838-6 


•26448 


7 


8 


42972 


•63318 


5055^6 


•70377 


2685.9 


•42909 


1828-8 


-26217 


8 


9 


38197 


.58203 


4982^3 


•69743 


2665-1 


•42571 


1819-1 


.25986 


9 


10 


34377 


-53627 


4911-2 


•69118 


2644.6 


•42235 


1809-6 


•25757 


10 


11 


31252 


4-49488 


4842-0 


3 •68502 


2624.4 


3-41903 


1800-1 


3 •25529 


11 


12 


28648 


•45709 


4774-7 


•67895 


2604-5 


•41572 


1790-7 


•25303 


12 


13 


26444 


.42233 


4709^3 


•67296 


2584-9 


•41245 


1781-5 


-25077 


13 


14 


24555 


•39014 


4845^7 


•66705 


2565.6 


•40919 


1772^3 


-24853 


14 


15 


22918 
21486 


•36018 


4583-8 


•66122 


2546-6 


•40597 
3.40276 


1763-2 


•24629 


15 


16 


4-33215 


4523-4 


3^65547 


2527-9 


1754-2 


3 • 24407 


16 


17 


20222 


•30582 


4464-7 


•64979 


2509-5 


•39958 


1745^3 


-24186 


17 


18 


19099 


•28100 


4407-5 


•64419 


2491-3 


•39642 


1736^5 


•23967 


18 


19 


18093 


•25752 


4351-7 


-63365 


2473-4 


.39329 


1727^8 


-23748 


19 


20 


17189 


•23524 


4297-3 


•63319 


2455-7 


•39017 


1719^1 


•23530 


20 


21 


16370 


4-21405 


4244-2 


3-62780 


2438-3 


3-38708 


1710^6 


3^23314 


21 


22 


15626 


•19385 


4192-5 


•62247 


2421-1 


•38401 


1702^1 


•23098 


22 


23 


14947 


•17454 


4142-0 


•61720 


2404-2 


•38097 


1693^7 


•22884 


23 


24 


14324 


•15606 


4092-7 


•61200 


2387.5 


-37794 


1685-4 


-22670 


24 


25 


13751 


•13833 


4044-5 


•60686 


2371-0 


•37494 


1677-2 


•22458 


25 


26 


13222 


4^12130 


3997-5 


3^60178 


2354-8 


3^37195 


1669 1 


3^22247 


26 


27 


12732 


•10491 


3951-5 


•59676 


2338-8 


-36899 


1661^0 


-22037 


27 


28 


12278 


•08911 


3906-6 


•59180 


2323-0 


-36604 


1653^0 


-21827 


28 


29 


11854 


•07387 


3862-7 


•58689 


2307.4 


-36312 


1645^1 


.21619 


29 


30 


11459 


•05915 
4^04491 


3819-8 


-58204 


2292.0 


•36021 


1637-3 


•21412 


30 


31 


11090 


3777-9 


3-57724 


2276.8 


335733 


1629-5 


3-21206 


31 


32 


10743 


•03112 


3736-8 


•57250 


2261.9 


•35446 


1621-8 


-21000 


32 


33 


10417 


•01776 


3698-6 


•56780 


2247-1 


-35162 


1614-2 


-20796 


33 


34 


10111 


4-00479 


3657-3 


•56316 


2232-5 


.34879 


1606-7 


-20593 


34 


35 


9822-2 


3-99221 
3-97997 


3618-8 


•55856 


2218-1 


•34598 


1599-2 


•20390 


35 


36 


9549-3 


3581-1 


3-55401 


2203-9 


3=34318 


1091-8 


3-20189 


36 


37 


9291-3 


•96807 


3544-2 


•54951 


2189-8 


•34041 


1584-5 


.19988 


37 


38 


9046-7 


•95649 


35^8-0 


•54506 


21760 


-33765 


1577^2 


.19789 


38 


39 


8814.8 


•94521 


3472-6 


•54065 


2162-3 


•33491 


1570^0 


-19500 


39 


40 


8594.4 


•93421 
3-92349 


3437-9 
3403-8 


•53829 


2148-8 


•33219 
3^32949 


1562-9 


•19392 


40 


41 


8384.8 


3-53197 


2135-4 


1555-8 


3-19195 


41 


42 


8185.2 


•91302 


3370-5 


•52769 


2122-3 


•32680 


1548-8 


.18999 


42 


43 


7994.8 


•90281 


3337-7 


•52345 


2109-2 


•32412 


1541^9 


.18804 


43 


44 


7813.1 


•89282 


3305-7 


.51925 


2096-4 


-32147 


1535^0 


•18610 


44 


45 


7639-5 


-88306 


3274-2 


•51510 


2083-7 


-31883 


152R-2 


.18417 


45 


46 


7473.4 


3-87352 


3243.3 


3 •51098 


2071-1 


3-31621 


1521-4 


3.18224 


46 


47 


7314.4 


•86418 


3213-0 


.50691 


2058-7 


.31360 


1514.7 


.18032 


47 


48 


7162.0 


•85503 


3183-2 


.50287 


2046.5 


.31101 


1508.1 


.17842 


48 


49 


7015.9 


•84608 


3154-0 


.49886 


2034-4 


-30843 


1501.5 


-17652 


49 


50 


6875-6 


-83731 


3125-4 


•49490 
3.49097 


2022-4 


•30587 
3.30332 


1495.0 


.17402 


50 


51 


6740-7 


3-82871 


3097-2 


2010.6 


1488-5 


3-17274 


51 


52 


6611-1 


•82027 


3069- 6 


.48707 


1998.9 


.30079 


1482.1 


-17087 


52 


53 


6488-4 


•81200 


3042-4 


•48321 


1987-3 


.29827 


1475.7 


.16900 


53 


54 


6368.3 


•80388 


3015-7 


.47939 


1975-9 


•29577 


1469.4 


.16714 


54 


55 


6250.5 
6138. 9 


•79591 


2989-5 


.47559 


1964-6 


.29328 
3-29081 


1463-2 


.16529 


55 


56 


3. 78809 


2963-7 


3.47183 


1953.5 


1457.0 


3.16344 


56 


57 


6031.2 


•78040 


2938.4 


.46811 


1942.4 


.28835 


1450.8 


.16161 


57 


58 


5927.2 


•77285 


2913.5 


.46441 


1931.5 


.28590 


1444.7 


.15978 


58 


59 


5826.8 


.73542 


2889. 


.46075 


1920.7 


.28347 


1438.7 


.1570B 


59 


60 


5729.6 


.75813 


2864.9 


.45711 


1910.1 


.28105 


1432.7 


.15615 


60 



512 











TABLE I.- 


-RADII OF CURVES. 










iDeg 

ilVlin 


4° 


5° 


6° 


7° 


Deg 


Radius. I 


.OgM 


Radius. 1 


.ogH 


Radius. 1 


.OgH 


Radius. L 


-og^ 


iViin 


! 


1432-7 3 


15615 


1146.3 3 


.05929 


955-37 2 


.98017 


819-02 2 


91329 





■! 1 


1426 


7 


15434 


1142 


5 


.05784 


952 


.72 


.97896 


817 


08 


91226 


1 


2 


1420 


8 


15255 


1138 


7 


.05640 


950 


-09 


•97776 


815 


14 


91123 


2 


1 3 


1415 





15078 


1134 


9 


05497 


947 


-48 


.97857 


813 


22 


91021 


3 


' 4 


1409 


2 


14897 


1131 


2 


05354 


944 


-88 


•97537 


811 


30 


90918 


4 


6 


1403 


5 


14720 


1127 


5 


.05211 
05069 


942 


-29 


.97418 


809 


40 


90816 


5 


1397 


8 3 


14543 


1123 


8 3 


939 


• 72 2 


97300 


807 


50 2 


90714 


6 


i 7 


1392 


1 


14367 


1120 


2 


04928 


937 


■ 18 


.97181 


805 


61 


90612 


7 


8 


1386 


5 


.14191 


1118 


5 


04787 


934 


.62 


.97063 


803 


73 


90511 


8 


' 9 


1380 


9 


.14017 


1112 


9 


04646 


932 


.09 


.96945 


801 


86 


90410 


9 


10 

J 11 

12 


1375 


4 


13843 


1109 


3 


04506 


929 


57 


96828 


800 


00 


90309 


10 


1339 


9 3 


13839 


1105 


8 3 


04366 


927 


07 2 


96711 


798 


14 2 


90208 


11 


1384 


5 


13497 


1102 


2 


04227 


924 


58 


96594 


796 


30 


90107 


12 


i 13 


1359 


1 


13325 


1098 


7 


04088 


922 


10 


•96478 


794 


46 


90007 


13 


14 


1353 


8 


.13154 


1095 


2 


03949 


919 


64 


9636. 


792 


63 


89907 


14 


15 

J 16 


1348 


4 


12983 


1091 


7 


03811 


917 


19 


96246 


790 


81 


89807 


15 


1343 


2 3 


.12813 


1088 


3 3 


03874 


914 


75 2 


96130 


789 


00 2 


89708 


16 


1 17 


1338 





.12644 


1084 


8 


03537 


912 


33 


96015 


787 


20 


89608 


17 


! 18 


1332 


8 


.12475 


1081 


4 


03400 


909 


92 


95900 


785 


41 


89509 


18 


' 19 


1327 


6 


12307 


1078 


1 


03264 


907 


52 


95785 


783 


62 


89410 


19 


20 
21 


1322 


5 


12140 


1074 


7 


03128 
02992 


905 


13 


95871 


781 


84 


89312 


20 


1317 


5 3 


11974 


1071 


3 3 


902 


76 2 


95557 


780 


07 2 


89213 


21 


22 


1312 


4 


11808 


1068 





02857 


900 


40 


95443 


778 


31 


89115 


22 


23 


1307 


4 


11642 


1064 


7 


02723 


898 


05 


95330 


776 


55 


89017 


23 


24 


1302 


5 


11477 


1061 


4 


02589 


895 


71 


95217 


774 


81 


88919 


24 


25 
' 26 


1297 


6 


11313 


1058 


2 


02455 


893 


39 


95104 


773 


07 


88821 
88724 


25 


1292 


7 3 


11150 


1054 


9 3 


02322 


891 


08 2 


94991 


771 


34 2 


26 


27 


1287 


9 


10987 


1051 


7 


02189 


888 


78 


94879 


769 


61 


88627 


27 


( 28 


1283 


1 


10825 


1048 


5 


02056 


886 


49 


94767 


767 


90 . 


88530 


28 


29 


1278 


3 


10663 


1045 


3 


01924 


884 


21 


94655 


766 


19 . 


88433 


29 


30 
31 


1273 


6 


10502 


1042 


1 


01792 


881 


95 


94544 


764 


49 


88337 


30 


1268 


9 3 


10341 


1039 


3 


01661 


879 


69 2 


94433 


762 


80 2. 


88241 


31 


32 


1264 


2 


10182 


1035 


9 


01530 


877 


45 


94322 


781 


11 


88145 


32 


33 


1259 


6 


10022 


1032 


8 


01400 


875 


22 


94212 


759 


43 . 


88049 


33 


: 34 


1255 





09864 


1029 


7 


01270 


873 


00 


94101 


757- 


76 . 


87953 


34 


' 35 
36 


1250 


4 


09705 


1026 


6 


01140 


870 


80 ! 


93991 


756- 


10 


87858 


35 


1245 


9 3 


09548 


1023 


5 3 


01010 


868 


60 2 


93882 


754- 


44 2- 


87762 


36 


37 


1241 


4 


09391 


1020 


5 


00882 


866 


41 


93772 


752. 


80 1 . 


87668 


37 


38 


1236 


9 


09234 


1017 


5 


00753 


864 


24 


93663 


751. 


16 1 . 


87573 


38 


39 


1232 


5 


09079 


1014 


5 


00625 


862 


07 


93554 


749. 


52 . 


87478 


39 


40 
41 


1228 


1 


08923 


1011 


5 


00497 


859 


92 


93446 


747- 


89 


87384 


40 


1223 


7 3 


08789 


1008 


6 3 


00370 


857 


78 ! 2 


93337 


746. 


27 2- 


87290 


41 


, 42 


1219 


4 


08614 


1005 


6 


00242 


855 


85 


93229 


744- 


66 i . 


87196 


42 


43 


1215 


1 


08461 


1002 


7 3 


00116 


853 


53 


93122 


743- 


06 1 . 


87102 


43 


44 


1210 


8 


08308 


999 


76 2 


99989 


851 


42 


93014 


741- 


46 . 


87008 


44 


1 45 
46 


1206 


6 


08155 


996 
993 


87 


99863 


849 


32 


92907 


739- 


86 


86915 


45 


1202 


4 3 


08003 


99 2 


99738 


847 


23 2 


92800 


738 


28 2- 


86822 


46 


47 


1198 


2 


07852 


991 


13 


99613 


845 


15 


92693 


736 


70 


86729 


47 


48 


1194 





07701 


988 


28 


99488 


843 


08 


92587 


735 


13 


86636 


48 


49 


1189 


9 


07550 


985 


45 


99363 


841 


02 


92480 


733 


56 


86544 


49 


50 
51 


1185 


8 


07400 


982 


64 . 


99239 


838 


97 


92374 


732 


01 


86451 


50 


1181 


7 I 3 


07251 


979 


84 2 


99115 


836 


93 2 


92289 


730 


45 2 


86359 


51 


1 52 


1177 


7 i 


07102 


977 


08 


98992 


834 


90 


92163 


728 


91 


86267 


52 


53 


1173 


6 


06954 


974 


29 


98889 


832 


89 


92058 


727 


37 


86175 


53 


54 


1169 


7 


08806 


971 


54 1 


98746 


830 


88 


91953 


725 


84 1 


86084 


54 


55 
56 


1165 


7 


06658 


988 


81 i 


98624 


828 


88 


91849 


724 


31 


85992 


55 


1161 


8 ; 3 


08511 


986 


09 1 2 


98501 


'826 


89 2 


91744 


722 


79 2 


85901 


56 


, f>7 


1157 


9 


06385 


983 


39 


98380 


824 


91 


91640 


721 


28 


.85810 


57 


; 58 


1154 





06219 


980 


70 


98258 


822 


93 


.91536 


719 


.77 


.8571§ 


58 


59 


1150 


1 


06074 


958 


03 


98137* 


820 


97 


.91433 


718 


.27 


.85629 


59 


60 

i 


1146 


3 


05929 


955.37 


98017 


819.02 


.91329 


716.78 


.85538 1 60 



513 













TABLE 


I. 


—RADII OF CURVES. 












Deg. 


8^ 


9° 


10° 


ir 


Deg 


Min. 


Radius. 


LogJJ 


Radius. 


LogU 


Radius. 


logM 


Radius. 


LogJ^ 


Min 



1 
2 
3 
4 
5 


716 
715 
713 
712 
710 
709 


78 

.29 
81 
34 
87 
40 


2 


85538 
.85448 
■85358 
85268 
85178 
85089 


637 
636 
634 
633 
632 
631 
630 
629 
627 
626 
625 


27 
10 
93 
76 
60 
44 

29 
14 
99 
85 
71 


2 


80432 
80352 
80272 
80192 
80113 
80033 


573 
572 
571 
570 
569 
568 


69 
73 
78 
84 
90 
96 


2 


75867 
75795 
75723 
75651 
75579 
755QJB 


521 
520 
520 
519 
518 
517 


67 
88 
10 
32 
54 
76 


2 


71739 
71674 
71608 
71543 
71478 
71413 


01 

1 

2 

3 

4 

5 


6 
7 
8 
9 
10 


707 
706 
705 
703 
702 


95 
49 
05 
61 
17 


2 


85000 
84911 
84822 
84733 
84644 


2 


79954 
79874 
79795 
79716 
79637 


568 
567 
566 
565 
564 


02 
09 
16 
23 
31 


2. 


75436 
75365 
75293 
75222 
75151 


516 
516 
515 
514 
513 


99 

21 
44 
68 
91 


2. 


71348 
71283 
71218 
71153 
71088 


6 
7 
8 
9 
10 


11 
12 
13 
14 
15 


700 
699 
697 
696 
695 


75 
33 
91 
50 
09 


2 


84556 
84468 
84380 
84292 
84204 


624 
623 
622 
621 
620 


58 

45 
32 
20 
09 


2 


79558 
79480 
79401 
79323 
79245 


563 
562 
561 
560 
559 


38 

47 
55 
64 
73 


2 


75080 
75009 
7493G 
74868 
74798 


513 
512 
511 
510 
510 


15 
38 
63 
87 
11 


2. 


71024 
70959 
70895 
70831 
70767 


11 
12 
13 
14 
15 


16 
17 
18 
19 
20 


693 
692 
690 
689 
688 


70 
30 
91 
53 
16 


2 


84117 
84029 
83942 
83855 
83768 


618 
617 
616 
615 
614 


97 
87 
76 
66 
56 


2 


79167 
79089 
79011 
78934 
78856 


558 
557 
557 
556 

555 


82 

92 
02 
12 
23 


2 


74727 
74657 
74587 
74517 
74447 


509 
508 
507 
507 
506 


36 
61 
86 
12 
38 


2. 


70702 
70638 
70575 
70511 
70447 


16 
17 
18 
19 
20 


21 
22 
23 
24 
25 


686 
685 
684 
682 
681 


78 

42 
06 
70 
35 


2 


83682 
83595 
83509 
83423 
83337 


613 
612 
611 
610 
609 


47 
38 
30 
21 
14 

06 
99 
93 
86 
80 


2 


78779 
787C2 
78625 
78548 
78471 


554 
553 
552 
551 
550 


34 
45 
56 
68 
80 


2 


74377 
74307 
74238 
74168 
74099 


505 
5C4 
504 
503 
502 


64 
90 
16 
42 
69 


2 


70383 
70320 
70257 
70193 
70130 


21 
22 
23 
24 
25 


26 
27 
28 
29 
30 


680 
678 
677 
676 
674 


01 
67 
34 
01 
69 


2 


83251 
83166 
83080 
82995 
82910 


608 
606 
605 
604 
603 


2 


78395 
78318 
78242 
78165 
78089 


549 
549 
548 

547 
546 


92 
05 
17 
30 
44 


2 


74030 
73961 
73892 
73823 
73754 


501 
501 
5C0 
499 
^f9 


96 
23 
51 
78 
06 


2 


70067 
70004 
69941 
69878 
69815 


26 
27 
28 
29 
30 


31 
32 
33 
34 
35 


673 
672 
670 
669 
668 


37 
06 
75 
45 
15 


2 


82825 
82740 
82656 
82571 
82487 


602 
601 
600 
599 
598 


75 
70 
65 
61 
57 


2 


78013 
77938 
77862 
77786 
77711 


545 
544 
543 
543 
542 


57 
71 
86 
CO 
15 


2 


73685 
73617 
73548 
73480 

73412 


4S8 
497 
496 
496 
495 


34 
62 
91 
19 
48 


2 


69752 
69690 
69627 
69565 
69503 


31 
32 
33 
34 
35 


36 
37 
38 
39 
40 


666- 

665 

664 

663 

661 


86 

57 
29 
01 
74 


2 


82403 
82319 
82235 
82152 
82068 


597 
596 
595 
594 
593 


53 
50 

47 
44 
42 


2 
7 


77636 
77561 
77486 
77411 
77336 


541 
540 
539 
538 
537 

537 
536 
535 

534 
c83 


30 
45 

• 61 
.76 

92 

• 09 
25 
42 
59 
77 


2 


73343 
.73275 
■73207 
.73140 

73072 


494 
484 
403 
492 
493 


77 
07 

• 36 

• 66 

• 96 


2 


69440 
•69378 
.69316 
.69254 
•69192 


36 
37 
38 
39 
40 


41 
42 
43 
44 
45 


660 

659 

657. 

656. 

655. 


47 
21 
95 
69 
45 


2 


81985 
81902 
81819 

81736 
81653 


592 
591 
590 
589 
588 


40 
38 
37 
36 
36 


77261 
77187 
77112 
77038 
76964 


2 


.7S0C4 
.72937 
•72869 
.72802 
•72735 


491 
480 
489 
480 
4?S 


• 26 

■ 56 

• 86 
.17 

■ 48 


2 


.69131 
•69069 
•69007 
•68946 
68884 


41 
42 
43 
44 
45 


46 
47 
48 
49 
50 


654. 
652. 
651. 
650. 
649 


20 
96 
73 
50 
9,1 


2 


81571 
81489 
81406 
81324 
81243 


587 
586 
585 
584 
583 


36 
36 
36 
37 
38 


2 


76890 
76816 
76742 
76669 
76595 


532 
532 
531 
530 
529 


94 
12 
30 
49 
f7 


2 


•72668 
.72601 
•72534 
•72467 
72401 


487 
487 
486 
485 
485 


■79 
.10 
.42 
.73 
.05 


2 


68823 
.68762 
•68701 
•68640 
.68579 


46 
47 
48 
49 
50 


51 
52 
53 
54 
55 


648 
646 
645 
644 
643 


05 
84 
63 

42 
9,9, 


2 


81161 
81079 
80998 
80917 
80836 


582 
581 
580 
579 
578 

577 
576 
575 
574 
573 


40 
42 
44 
47 
49 

53 
56 
60 
64 
69 


2 


76522 
76449 
76376 
76303 
76230 


528 
528 
527 
526 
525 


86 
05 
25 
44 
64 


2 


72334 
72267 
722CI 
72135 
72069 


484 
483 
483 
482 


• 37 

■ 69 

02 

34 

e7 


2 


•68518 
•68457 
•68386 
•68335 
68275 


51 
52 
53 
54 
55 


56 
57 
58 
59 
60 


642 
640 
639 
638 
637 


02 
83 
64 
45 
27 


y 


80755 
80674 
80593 
80513 
80432 


2 


76157 

76084 

76012 

•75939 

•75867 


524 
524 
523 
522 
521 


84 
05 
25 
46 
67 


2 


720C3 
71937 
71871 
71805 
71739 


4ca 

480 
479 
479 
478 


CO 
33 
67 
00 
34 


2 


68214 
68154 
68004 
eCC23 
67973 


56 
57 
58 
59 
60 



514 



1 










TABLE I. 


—RADII OF CURVES. 










Deg. 

1 


Radius. 


LogJJ 


Deg. 


Radius. 


Log JJ 


Deg. 


Radius. 


LogJJ 


Deg. 

21° 

10 
20 
30 
40 
50 

22° 
10 
20 
30 
40 
50 


Radius 


Log-K 


12° 

\ 
6 
8 


478 

477 
475 
474 
473 


34 
02 
71 
40 
10 


2 


67973 
67853 
67734 
67614 
67495 


14° 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 
22 
24 
26 
28 
30 
32 
34 
36 
38 
40 
42 
44 
46 
48 

50 
52 
54 
56 
58 

15° 

2 
4 
6 
8 

10 
12 
14 
16 
18 

20 
22 
24 
26 
28 

30 
32 
34 
33 
38 

4G 
42 
44 
46 
48 

50 
52 
54 
56 
58 

16° 


410 
409 
408 
407 
406 

405 
404 
403 
402 
401 


28 

31 
34 
38 
42 

47 
53 
53 
65 
71 


2 


.61307 
61205 
61102 
61000 
60898 


16° 

5 
10 
15 
20 
25 

30 
35 
40 
45 
50 
55 

17° 

5 
10 
15 
20 
25 

30 
35 
40 
45 
50 
55 

1S° 
5 
10 
15 
20 
25 

30 
35 
40 
45 
50 
55 

19° 

5 
10 
15 
20 

25 

30 
35 
40 
45 
50 
55 

30° 

5 
10 
15 
20 

25 

30 
35 
40 
45 
50 
55 

21° 


359 

357 
355 
353 
351 
350 


.26 
42 
59 
77 
98 
21 


2 


.55541 
.55317 
•55094 
.54872 
.54652 
■54432 


274 
272 
270 
268 
266 
264 


37 

■ 23 
.13 

• 06 

• 02 
02 


2.43833 
•43494 
.43157 
.42823 
.42492 


10 


471 
470 
469 
467 
466 


81 
53 
25 
98 
72 


2 


67376 
67258 
67140 
67022 
66905 


2 


60796 
60694 
60593 
60492 
60391 


•42163 


12 
14 
16 
18 


348 
346 

344 
343 
341 
339 


45 
71 
99 
29 
60 
93 


2 


.54214 
.53997 
53780 
53565 
53351 
53138 


262 
260 
258 
256 
254 
252 


04 
10 
18 
29 
43 
60 


2.41837 
•41513 
■41192 
•40873 


20 
22 


465 
464 
462 
461 
460 


46 
21 
97 
73 
50 


2 


66788 
66671 
66555 
66439 
66323 


400 
399 
398 
398 
397 


78 
86 
94 
02 
11 


2 


60291 
60190 
60090 
59990 
59891 


•40557 
.40243 


24 
26 
28 


338 
336 
335 
333 
331 
330 

328 

327 

325. 

324. 

322. 

321. 


27 
64 
01 
41 
82 
24 

68 
13 
60 
09 
59 
10 


2 


52927 
52716 
52506 
52297 
5209C 
5188S 


23° 

10 
20 
30 
40 
50 

24° 
10 
20 
30 
40 
50 

25° 

3p 
26° 

30 

27° 

3? 

28° 

30 

29° 

30 
30° 

30 

31° 

32 

33 

34 

35 

36 
37 

38 
39 
40 

41 
42 
43 
44 
45 

46 
47 

48 
49 
50 

52 
54 
56 
58 
60 


250 
249 
247 
245 
243 
242 


79 
01 
26 
53 
82 
14 


2.39931 
39622 
39315 


30 
32 
34 


459 
458 
456 
455 

454 

453 
452 
450 
449 
448 


28 
06 
85 
65 

45 

26 
07 
89 
72 
56 


2 


66207 
66092 
65977 
65863 
65748 


396 
395 
394 
393 
392 

391. 

390. 

389. 

389 

388 


20 
30 
40 
50 
61 
72 
84 
96 
08 
21 


2 


59791 
59692 
59593 
59494 
59398 


.39015 
38707 
38407 


36 
38 


2 


51677 
51472 
51269 
51066 
50864 
50663 


240 

238. 

237. 

235. 

234. 

232. 

231. 
226. 
222. 
218- 


49 
85 
24 
65 
08 
54 

01 
55 
27 
15 


2.38109 
37813 


40 
42 
44 
46 


2 


65634 
65521 
65407 
65294 
65181 


2. 


59298 
59199 
59102 
59004 
58907 


.37519 

.37227 

36937 

.36649 


48 


319 

318. 

316. 

315 

313. 

312 

311. 

309 

308. 

306. 

305. 

304. 

302. 

301. 

300. 

299 

297 

296 

295 

294 
292 
291 
290 
289 


62 
16 
71 
28 
86 
45 

06 
67 
30 
95 
60 
27 
94 
63 
33 
04 
77 
50 

25 
00 
77 
55 
33 
13 


2 


50464 
50265 
50067 
49869 
4967b 
49478 


2-36363 


50 
52 
54 


447 
446 
445 
443 
442 


40 
24 
09 
95 
81 


2 


65069 
64957 
64845 
64733 
64622 


387 

386 

385- 

384 

383 


34 
48 
62 
77 
91 


2 


58809 
58713 
58616 
58519 
58423 


.35517 
•34688 
.33875 


56 
58 


214. 
210. 
206- 
203. 


18 
36 
68 
13 


2.33078 
•32296 


13° 

2 


441 

440 

439 

438 

43X 

436 

435 

433 

432 

431 


68 
56 
44 
33 

22 

12 
02 
93 
84 
76 


2 
2 


64511 
64400 
64290 
64180 
64070 

63960 
63851 
63742 
63633 
63524 


383. 

382 

381. 

380. 

379. 


06 
22 
38 
54 
71 


2 


58327 
58231 
58135 
58040 
57945 


2 


49234 
49090 
48898 
48706 
48515 
48325 


•31529 
•30776 


4 
6 
8 


199. 
196. 
193. 
190. 

187. 
181. 
176. 
171. 
166. 

161. 
157. 
153. 
149. 
146. 


70 
38 
19 
09 
10 
40 
05 
02 
28 
80 
58 
58 
79 
19 


2.30037 
•29310 
.28597 


10 
12 
14 
16 
18 


378 
378 
377 
376 
375 


88 

05 
23 
41 
60 


2 


57850 
57755 
57661 
57566 
57472 


27896 


2 


48136 
4794S 
4776C 
47573 
47388 
47203 


2.27207 
•25863 
•24563 
.23303 


20 


430 
429 
428 
427 
426 

425 
424 
423 
422 
421 


69 
62 
56 
50 
44 
40 
35 
32 
28 
26 


2 


63416 
63308 
63201 
63093 
62936 


374 
373 
373 
372 
371 


79 
98 
17 
37 
57 


2 


57378 
57284 
57191 
57097 

57004 


.22083 


22 
24 
26 
28 


2.20899 
.19749 
.18633 
•17547 


2 


47018 
46835 
46652 
46471 
46289 
46109 


30 


2 


62879 
62773 
62666 
62560 
62454 


370 
369 
369 
368 
367 

366 
366 
365 
364 
363 

363 
362 
361 
360 
360 

359 


78 
99 

20 
42 
64 

"86 
09 
31 
55 
78 

02 
26 
51 
76 
01 

26 


2 


56911 
56819 

56728 
56634 
56542 


•16492 


32 
34 
36 
38 


142. 

139. 

136 

133 

130- 

127 
125 
122 
120 
118 


77 
52 
43 
47 
66 
97 
39 
93 
57 
31 


2^15464 
• 14464 
•13489 
.12539 


287 
286 
285 
284 
283 
282 


94 
76 
58 

42 
27 
12 


2 


45830 
45751 
45573 
45396 
45219 
45044 


40 


420 
419 
418 
417 
416 

415" 

414 

413 

412 

411 


23 
22 
20 
19 
19 

19 
20 
21 
23 
25 


2 


62349 
62243 
62138 
62034 
61929 


2 


56450 
56358 
56266 
56175 
56084 


.11613 


42 
44 
46 
48 


2.10709 
.09827 
.08965 
.08124 


280 
279 
278 
277 
276 
275 


99 
86 
79 
64 
54 
45 


2 


44869 
44694 
44521 
44348 
44176 
44004 


50 


2 


61825 
61721 
61617 
61514 
61410 


2 


55993 
55902 
55812 
55721 
55631 


.07302 


52 
54 
56 
58 


114 
110 
106 
103 
100 


06 
13 
50 
13 
00 


2.05713 
.04192 
•02736 
•01340 


274 


37 


2 


43833 

1 


14° 


410 


28 


2 


61307 


2 


55541 


2^00000 



515 



TABLE II.— TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS 

















FOR A 1 


^ CURVE. 










A 


■^^^S- 


Ext. 
Dist. 


Chord 


A 


T^ng- 


Ext. 
Dist. 

E. 


Long 
Chore 


A 

3 21° 

9 10 
5 20 

1 30 

7 40 

2 50 

8 22" 

4 10 
C 20 

5 30 

1 40 

7 50 

2 23° 

8 10 

3 20 

9 30 

4 40 

50 

5 24° 

1 10 
B 20 

2 30 
7 40 

2 50 

7 25° 

3 10 
B 20 
3 30 
B 40 
3 50 

5 26° 
3 10 
3 20 
J 30 
3 40 
J 50 

J 27° 
J 10 
J 20 
J 30 
1 40 
? 50 

5 28° 
. 10 
) 20 
) 30 
t 40 

> 50 

! 29° 

1 10 

> 20 
) 30 
. 40 
) 50 

»30° 

\ 10 
^ 20 
. 30 
) 40 
50 

31° 


Tajlg. 


JE. 

) 97.58 
5 99.15 
\ 100-75 
J 102.35 
1103.97 
L 105.60 


Lotiff 


1° 

10' 
20 
30 
40 
50 


50 
58 
66 
75 
83 
91 


.00 
.34 
• 67 
.01 
.34 
.68 










• 218 
.297 

388 
.491 
.606 

• 733 


100 
116 
133 
150 
166 
183 


• OC 

• 67 

• 33 
.00 
.66 

33 


11° 

10 
20 
30 
40 
50 

12° 
10 
20 
30 
40 
50 

13° 
10 
20 
30 
40 
50 

14° 

10 
20 
30 
40 
50 

15° 

10 
20 
30 
40 
50 

16° 

10 
20 
30 
40 
50 

17° 
10 
20 
30 
40 
50 

18° 
10 
20 
30 
40 
50 

19° 

10 
20 
30 
40 
50 

^0° 

10 
20 
30 
40 
50 

31° 


551 
560 
568 
576 
585 
593 


.70 
.11 

• 53 
.95 
.36 

• 79 


26 
27 
28 
28 
29 
30 


• 500 
.313 

• 137 
.974 
.824 

686 


1098. 
1114. 
1131. 
1148. 
1164. 
1181. 


1061.1 
1070. ( 
1079.: 
1087. i 
1096.^ 
1105.] 


2088.3 
2104.7 
2121.1 
2137.4 
2153.8 
2170.2 


3° 
10 
20 
30 
40 
50 


100 
108 
116 
125 
133 
141 


• 01 
.35 
68 
02 
36 
70 





.873 
.024 
188 
364 
552 
752 


199 
216 
233 
249 
266 
283 


.99 
.66 
32 
98 
65 
31 


602 
610 
619 
627 
635 
644 


■ 21 
.64 
.07 
.5C 
•93 
.37 


31 
32 
33 
34 
35 
36 


.561 
.447 
347 
.259 
.183 
.120 


1197. 
1214. 
1231. 
1247. 
1264. 
1280. 


1113.' 
1122.-^ 
1131. ( 
1139.' 
1148-^ 
1157. ( 


ri07.24 
L 108.90 
) 110-57 
1 112-25 
H13.95 
) 115.66 


2186.5 
2202-9 
2219-2 
2235-6 
2251-9 
2268.3 


3° 
10 
20 
30 
40 
50 


150 
158 
166 
175 
183 
191 


04 
38 
72 
06 
40 
74 


1 

2 
2 
2 
2 
3 


964 
188 
425 
674 
934 
207 


299 
316 
333 
349 
366 
383 


97 
63 
29 
95 
61 
27 


652 
661 
669 
678 
686 
695 


81 
.25 
.70 
.15 
■ 60 

06 


37 
38 
39 
39 
40 
42 


.069 

• 031 
.006 
.993 

• 992 

• 004 


1297. 
1313. 
1330. 
1346. 
1363. 
1380. 


1165.^ 
1174.^ 
1183.] 
1191. { 
1200. e 
1209. S 


ril7-38 
H19-12 
L 120-87 
\ 122.63 
) 124.41 
} 126-20 


2284.6 
2301.0 
2317.3 
2333.6 
2349.9 
2366.2 


4° 
10 
20 
30 
40 
50 


200 
208 
216 
225 
233 
241 


08 
43 
77 
12 
47 
81 


3 
3 

4 
4 
4 
5 


492 
790 
099 
421 
755 
100 


399 
416 
433 
449 
466 
483 


92 
58 
24 
89 
54 
20 


703 
711 
720 
728 
737 
745 


.51 
.97 
44 
90 
37 
85 


43 
44 
45 
46 
47 
48 


.029 
.066 
.116 
.178 
.253 
• 341 


1396. 
1413. 
1429. 
1446. 
1462. 
1479. 


1217. £ 
1226. € 
1235. S 
1244. C 
1252. J 
1261. e 


1 128-00 
) 129-82 
1 131-65 
) 133-50 
1 135-36 
) 137.23 


2382.5 
2398.8 
2415.1 
2431.4 
2447.7 
2464.0 


5° 
10 
20 
30 
40 
50 


250 
258 
266 
275 
283 
291 


16 
51 
86 
21 
57 
92 


5 
5 
6 
6 
7 
7 


459 
829 
211 
606 
013 
432 


499 
516 
533 
549 
566 
583 


85 
5C 
15 
80 
44 
09 


754 
762 
771 
779 
788 
796 


32 
80 
29 
77 
26 
75 


49 
50 
51 
52 
53 
55 


441 
554 
679 
818 
969 
132 


1495. 
1512. 
1528. 
1545. 
1561. 
1578. 


1270.5 
1279. C 
1287.7 
1296. E 
1305.8 
1314-C 


' 139.11 

1 141.01 

' 142-93 

144-85 

146-79 

148.75 


2480.2 
2496.5 
2512.8 
2529.0 
2545.3 
2561.5 


6° 

10 
20 
30 
40 
50 


300 

308 

316 

325 

333. 

342 


28 
64 
99 
35 
71 
08 


7 
8 
8 
9 
9 
10 


863 
307 
762 
230 
710 
202 


599 

616 

633 

649 

666. 

682. 


73 
38 
02 
66 
30 
94 


805 
813 
822 
830 
839 
847 


25 
75 
25( 
76 
27 
78 


56 
57 
58 
59 
61 
62 


309 
498 
699 
914 
141 
381 


1594.1 
1611. J 
1627.1 
1644.: 
1660.1 
1677. J 


1322.fi 
1331.6 
1340.4 
1349.2 
1358-0 
1366.8 
1375.6 
1384.4 
1393.2 
1402.0 
1410-9 
1419.7 


150.71 
152.69 
154.69 
156-70 
158.72 
160.76 


2577.8 
2594.0 
2610.3 
2626.5 
2642.7 
2658.9 


r 

10 
20 
30 
40 
50 


350. 
358. 
367. 
375- 
383. 
392. 


44 
81 
17 
54 
91 
28 


10. 
11. 
11. 
12. 
12. 
13. 


707 
224 
753 
294 
847 
413 


699. 
716. 
732. 
749. 
766- 
782. 


57 
21 
84 
47 
10 
73 


856 

864 

873. 

881. 

890. 

898. 


30 
82 
35 
88 
41 
95 


63 

64 

66. 

67. 

68. 

70. 


634 
900 
178 
470 
774 
091 


1693. { 
1710.; 
1726. i 
1743. i 
1759.' 
1776.:; 


162.81 
164.87 
166.95 
169-04 
171.15 
173.27 


2675.1 
2691.3 
2707.5 
2723.7 
2739.9 
2756.1 


8° 
10 
20 
30 
40 
50 


400. 
409. 
417. 
425. 
434. 
442. 


66 
03 
41 
79 
17 
55 


13. 
14- 
15. 
15. 
16- 
17. 


991 
582 
184 
799 
426 
066 


799. 
815. 
832. 
849. 
865- 
882. 


36 
99 
61 
23 
85 
47 


907. 
916- 
924. 
933. 
941. 
950. 


49 
03 
58 
13 
69 
25 


71. 
72. 
74. 
75. 
76. 
78. 


421 
764 
119 
488 
869 
264 


1792. e 
1809.] 
1825. E 
1842. C 
1858.^ 
1874. c 


1428.6 
1437.4 
1446-3 
1455-1 
1464.0 
1472-9 


175.41 
177.55 
179.72 
181.89 
18408 
186.29 


2772.3 
2788.4 
2804.6 
2820.7 
2836.9 
2853.0 


9° 

10 
20 
30 
40 
50 


450. 
459. 
467. 
476. 
484. 
492. 


93 
32 
71 
10 
49 
88 


17. 
18. 
19. 
19. 
20. 
21. 


717 
381 
058 
74e 
447 
161 


899- 
915. 
932. 
948. 
965. 
982. 


09 
7G 
31 
92 
53 
14 


958. 
967. 
975. 
984. 
993- 
1001. 


81 
38 
96 
53 

12 
70 


79. 
81. 
82. 
83. 
85. 
86. 


671 
092 
525 
972 
431 
904 


1891.2 
1907.£ 
1924.2 
1940. f 
1957.] 
1973. e 


1481. 8 
1490-7 
1499. 6 
1508.5 
1517.4 
1526.3 


188.51 
190.74 
192.99 
195-25 
197-53 
199-82 


2869.2 
2885-3 
2901.4 
2917.6 
2933.7 
2949-8 


10° 

10 
20 
30 
40 
50 


501 
509 
518 
526 
534 
543 


28 
68 
08 
48 
89 
29 


21. 

22 

23 

24 

24 

25 


886 
624 
375 
138 
913 
700 


998. 
1015. 
1031. 
1048. 
1065 
1081 


74 
35 
95 
54 
14 
73 


1010 

1018. 

1027 

1036. 

1044. 

1053 


29 
89 
49 
09 
70 
31 


88 

89 

91 

92. 

94. 

96 


389 
888 
399 
924 
462 
013 


1989. c 
2006. £ 
2022.'; 
2039.] 
2055. £ 
2071. c 

2088. S 


1535.3 
1544.2 
1553-1 
1562.1 
1571.0 
1580.0 


202.12 
204.44 
206.77 
209.12 
211.48 
213-86 


2965-9 
2982-0 
2998.1 
3014.2 
3030.2 
3046.3 


111 


551 


.70 


26.500 


1098.33 


1061 


93 


97. 


577 


1589.0 


216.25 


3062.4 



516 



TABLE II.— TANGENTS, 



EXTERNAL DISTANCES, 
FOR A 1° CURVE. 



AND LONG CHORDS 



A 


"•"^^"^S 


• 


Ext. 
DIst. 


Long 
Chord 
xc\ 


A 


Tang. 


Ext. 
Dist. 
X. 


ChSixi 
xc. 


A 


Tang 


• 


Ext. 
Dist. 
X. 


Long 
Chord 

xc. 


31° 

10' 
20 
30 
40 
50 

10 
20 
80 
40 
50 

33; 

in 
20 

80 
40 
50 

34° 

: 10 
20 

30 

40 
50 

35° 

; 10 

20 
30 
40 
50 

36° 

' 20 
30 
40 
50 

37° 

10 
20 
30 
40 
50 

38° 

10 

: 20 

^ 30 

40 

50 

39° 

10 
20 
30 
i .40 
50 

40° 

10 
1 20 
30 
40 
50 

i4l° 


1589 
1598 
1606 
1615 
1624 
1633 




9 
9 
9 
9 


216 
218 

221 
223 
225 
228 


25 
66 
08 

51 
96 

42 


3062 
3078 
3094 
3110 
3126 
3142 


4 
4 
5 
5 
6 
6 


41° 

10 
20 
30 
40 
50 

42° 

10 
20 
30 
40 
50 

43^ 

10 
20 
30 
40 
50 

44° 

10 
20 
30 
40 
50 

45° 

10 
20 
30 
40 
50 

46° 

10 
20 
30 
40 
50 


2142.2 
2151.7 
2161.2 
2170-8 
2180. 3 
2189-9 


387 
390 
394 
397 
400 
404 


38 4013 
71 4028 
06 4044 
43 4059 
82 4075 
22 4091 


1 
7 
3 
9 
5 
1 


51° 

10 
20 
30 
40 
50 

52° 

10 
20 
30 
40 
50 

53° 

10 
20 
30 
40 
50 

54° 

10 
20 
30 
40 
50 

55° 
10 
20 
30 
40 
50 

56° 

10 
20 
30 
40 
50 


2732 
2743 
2753 
2763 
2773 
2784 


9 618 
11622 
4|627 
7|631 
.91636 
2|640 


39 4933 
814948 
244963 
694978 
16,4993 
66|5008 


4 
4 
4 
4 
4 
4 


1643 
1852 
1661 
1670 
1679 
1688 






1 
1 


230 
233 
235 
238 
240 
243 


90 
39 
90 
43 
96 
52 


3158 
3174 
3190 
3206 
3222 
3238 


6 
6 
6 
6 
6 
6 

6 
6 
6 
5 
5 
4 


2199.4 
2209.0 
2218.6 
2228.1 
2237-7 
2247-8 


407 
411 
414 
417 
421 
424 


64 4106 
07 4122 
52 4137 
99 4153 
48 4168 
98 4184 


6 
2 
7 
3 
8 
3 


2794 
2804 
2815 
2825 
2835 
2846 


5l645 
9 649 

2 654 
6 658 
9 663 

3 668 


17 5023 
70 5038 
25 5053 
83 5068 
42 5083 
03 5098 


4 
4 
4 
3 
3 
2 


1697 
1706 
1715 
1724 
1733 
1742 


2 
3 
3 
4 
5 
6 


246 
248 
251 
253 
256 
259 


08 
66 
26 
87 
50 
14 


3254 
3270 
3286 
3302 
3318 
3334 


2257-0 
2266-6 
2276-2 
2285-9 
2295-6 
2305-2 


428 
432 
435 
439 
442 
446 


50 4199 
04 4215 
59 4230 
16 4246 
75 4261 
35 4277 


8 
3 
8 
3 
8 
3 


2856 
2867 
2877 
2888 
2898 
2908 


7 672 
1 677 
5 681 
686 
4 691 
9 696 


685113 
32 5128 
99 5142 
68 5157 
40 5172 
13 5187 


1 

9 
8 
7 
6 


1751 
1760 
1770 
1779 
1788 
1797 


7 
8 


1 
2 
4 


261 
264 
267 
269 
272 
275 


80 
47 
16 
86 
58 
31 


3350 
3366 
3382 
3398 
3414 
3430 


4 
3 
2 
2 

1 



2314-9 
2324-6 
2334-3 
2344-1 
2353- 8 
2363-5 


449 
453 
457 
460 
464 
468 


98 4292 
62 4308 
27 4323 
954339 
64 4354 
35 4369 


7 
2 
6 

5 
9 


2919 
2929 
2940 
2951 
2961 
2972 


4 
9 
4 

5 
1 


700 
705 
710 
715 
720 
724 


89 
66 
46 
28 
11 
97 


5202 
5217 
5232 
5246 
5261 
5276 


4 
3 
1 
9 
7 
5 


1808 
1815 
1824 
1834 
1843 
1852 


6 
7 
9 

1 
3 
5 


278 
280 
283 
286 
289 
292 


05 
82 
60 
39 

20 
02 


3445 
3461 
3477 
3493 
3509 
3525 


9 
8 
7 
5 
4 
3 


2373-3 
2383-1 
2392-8 
2402-6 
2412-4 
2422-3 


472 
475 
479 
483 
487 
490 


08 4385 
82 4400 
59 4416 
374431 
16 4446 
98 4462 


3 

7 
1 
4 
8 
2 


2982 
2993 
3003 
3014 
3025 
3035 


7 
3 
9 
5 
2 
8 


729 
734 
739 
744 
749 
754 


85 
76 
68 
62 
59 
57 


5291 
5306 
5320 
5335 
5350 
5365 


3 

1 
9 
6 
4 

1 


1861 
1870 
1880 
1889 
1898 
1907 


6 
9 


294 
297 
300 
303 
306 
309 


86 
72 
59 
47 
37 
29 


3541 
3557 
3572 
3588 
3604 
3620 


1 

8 
6 
5 
3 


2432-1 
2441-9 
2451-8 
2461.7 
2471-5 
2481-4 


494 
498 
502 
506 
510 
514 


82 4477 
67 4492 
54 4508 
42 4523 
33 4538 
25 4554 


5 
8 

2 
5 
8 

1 


3046 
3057 
3067 
3078 
3089 
3100 


5 
2 
9 
7 
4 
2 


759 
764 
769 
774 
779 
784 


58 
61 
66 
73 
83 
94 


5379 
5394 
5409 
5423 
5438 
5453 


8 

5 
2 
9 
6 
3 


1917 
1926 
1935 
1945 
1954 
1963 


1 


3 
6 


312 
315 
318 
321 
324 
327 


22 
17 
13 
11 
11 
12 


3636 
3651 
3667 
3683 
3699 
3715 


1 
9 
7 
5 
3 



47° 
10 
20 
30 
40 
50 

48° 
10 
20 
30 
40 
50 

49° 

10 
20 
30 
40 
50 

50° 

10 
20 
30 
40 
50 

51" 


2491-3 
2501-2 
2511-2 
2521.1 
2531-1 
2541-0 


518 
522 
526 
530 
534 
538 


20 
16 
13 
13 
15 
18 


4569 
4584 
4599 
4615 
4630 
4645 


4 
7 
9 
2 
4 
7 


57° 
10 
20 
30 
40 
50 

58° 
10 
20 
30 
40 
50 


3110 
3121 
3132 
3143 
3154 
3165 


9 
7 
6 
4 
2 
1 


790 
795 
800 
805 
810 
816 


08 
24 
42 
62 
85 
10 


5467 
5482 
5497 
5511 
5526 
5541 


9 

5 
2 
8 
4 



1972 
1982 
1991 
2000 
2010 
2019 
2029 
2038 
2047 
2057 
2066 
2076 


9 
2 
5 
9 
2 
6 

4 
8 
2 
6 



330 
333 
336 
339 
342 
345 


15 

It 

32 
41 
52 


3730 
3746 
3762 
3778 
3793 
3809 


8 
5 
3 

8 
5 


2551-0 
2561-0 
2571-0 
2581-0 
2591-1 
2601-1 


542 
546 
550 
554 
558 
562 


23 
30 
39 
50 
63 
77 


4660 
4676 
4691 
4706 
4721 
4736 


9 
1 
3 
5 
7 
9 


3176 
3186 
3197 
3208 
3219 
3230 




• 

7 
7 

7 

I 
I 

9 


821 
826 
831 
837 
842 
848 


37 
66 
98 
31 
67 
06 


5555 
5570 
5584 
5599 
5613 
5628 


6 
2 
7 
3 
8 
3 


348 

351 
§54 
358 
361 
384 


64 
78 
94 
11 
29 
50 


3825 
3840 
3856 
3872 
3888 
3903 


2 
9 
6 

I 

6 


2611-2 
2621.2 
2631-3 
9«41.4 
2651-5 
2661-6 


566 
571 
575 
579 
583 
588 


94 
12 
32 
54 
78 
04 


4752 
4767 
4782 
4797 
4812 
4827 


1 
3 
4 
5 
7 
8 


59° 

10 
20 
30 
40 
50 

60° 

10 
20 
30 
40 
50 

61° 


3241 
3252. 
3263" 
3274 
3285 
3296 


853 
858 
864 
869 
875 
880 


4615642 
8915657 
3415671 
82J5686 
3215700 
8415715 


8 
3 
8 
3 
8 
2 


2085 
2094 
2104 
2113 
2123 
2132 


4 
9 
3 
8 

3 

7 


367 
370 
374 
377 
380 
384 


72 
95 
20 
47 
76 
06 


3919 
3935 
3950 
3966 
3981 
3997 


3 


t 

9 

5 


2671-8 
2681-9 
2692-1 
2702-3 
2712-5 
2722-7 


592 
596 
600 
605 
609 
614 


32 
62 
93 
27 
62 
00 


4842 
4858 
4873 
4888 
4903 
4918 


9 


1 
2 
2 
3 


3308 
3319 
3330 
3341 
3352 
3363 



1 
3 
4 
• 6 
8 


886 
891 
897 
903 
908 
914 


38 
95 
54 
15 
79 
45 


5729 
5744 
5758 
5772 
5787 
5801 


.7 
.1 
.5 
.9 
.3 
7 


2142 


2 


387 


• 38 


4013 


_1 


2732-9 


618 


39 


4933 


J 


3375 


■A 


920 


14 


58]6 


-0 



517 



TABLE II.— TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS 

FOR A 1° CURVE. 



A 


T^g. 


Ext. 
Dist. 


Long 
Chord 


A 


Tar^g- 


Ext. 
Dist. 


Long 
Chorcl 


A 


T^g. 


Ext. 
Dist. 


Long '' 
Chord ; 




JbJ, 


LC. 




jK. 


LC. 


81° 


JE. 




61° 


3375.0 


920-14 


5816. C 


71° 


4086-9 


1308.2 


6654-4 


4893-6 


1805-3 


7442.2 i 


10' 3386. 


3 


925 


85 


5830-4 


10 


4099- 


5 


1315- 


5 


6668- 





10 


4908- 





1814. 


7 


7454. 


9 1; 


20 3397- 


5 


931 


58 


5844-7 


20 


4112. 


1 


1322- 


9 


6681- 


6 


20 


4922- 


5 


1824. 


1 


7467. 


5 


30 13408- 


8 


937 


34, 


5859-1 


30 


4124- 


8 


1330- 


3 


6695- 


1 


30 


4937- 





1833. 


6 


7480. 


2 


40 3420. 


1 


943 


12! 


5873-4 


40 


4137- 


4 


1337- 


7 


6708- 


6 


40 


4951- 


5 


1843. 


1 


7492. 


8 


f^O 


3431. 


4 


948 


92j 


5887-7 


50 


4150- 


1 


1345- 


1 


6722. 


X 


50 

82° 


4966- 


1 


1852. 


6 


7505. 


4 ] 


62" 


3442. 


7 


954. 


75 


5902-0 


73° 


4162- 


8 


1352- 


6 


6735- 


6 


4980- 


7 


1862. 


2 


7518. 


3 


10 


3454. 


1 


960- 


60 


5916-3 


10 


4175. 


6 


1360. 


1 


6749. 


1 


10 


4995- 


4 


1871. 


8 


7530. 


5 


20 


3465. 


4 


966- 


48 


5930.5 


20 


4188- 


4 


1367. 


6 


6762. 


5 


20 


5010. 





1881. 


5 


7543. 


1 


30 


3476. 


8 


972- 


39 


5944.8 


30 


4201. 


2 


1375. 


2 


6776. 





30 


5024. 


8 


1891. 


2 


7555. 


6 


40 


3488. 


2 


978. 


31 


5959. C 


40 


4214- 





1382. 


8 


6789. 


4 


40 


5039. 


5 


1900. 


9 


7568. 


2 


50 


3499. 


7 


984. 


27 


5973-3 


50 


4226- 


8 


1390. 


4 


6802- 


8 


50 

83° 


5054. 


3 


1910. 


7 


7580. 


7 


63 


3511. 


1 


990. 


24 


5987-5 


73° 


4239- 


7 


1398. 





6816- 


3 


5069. 


2 


1920. 


5 


7593. 


2 \ 


10 


3522. 


6 


996. 


24 


6001-7 


10 


4252- 


6 


1405. 


7 


6829. 


6 


10 


5084. 





1930. 


4 


76(T5. 


6 [ 


20 


3534. 


1 


1002. 


3 


6015-9 


20 


4265- 


6 


1413. 


5 


6843- 


C 


20 


5099. 





1940. 


3 


7618. 


1 3 


30 


3545. 


6 


1008. 


3 


6030-0 


30 


4278- 


5 


1421 


2 


6856- 


4 


30 


5113. 


9 


1950- 


3 


7630. 


5 1 


40 


3557. 


2 


1014. 


4 


6044-2 


40 


4291- 


5 


1429 





6869. 


7 


40 


5128. 


9 


1960. 


2 


7643. 





50 


3568- 


7 


1020. 


5 


6058-4 


50 

74° 


4304 


6 


1436 


8 


6883. 


1 


50 

84° 


5143. 


9 


1970. 


3 


7655. 


4 - 


64^ 


3580. 


3 


1026- 


6 


6072-5 


4317 


6 


1444 


6 


6896 


4 


5159. 





1980- 


4 7667. 


8 


10 


3591. 


9 


1032- 


8 


6086-6 


10 


4330 


7 


1452 


5 


6909. 


7 


10 


5174- 


1 


1990. 


5 


7680. 


1 fl 


20 


3603. 


5 


1039 





6100-7 


20 


4343 


8 


1460 


4 


6923 





20 


5189- 


3 


2000. 


6 


7692. 


5 \ 


30 


3615- 


1 


1045 


2 


6114-8 


30 


4356 


9 


1468 


4 


6936 


2 


30 


5204- 


4 


2010. 


8 


7704. 


9 1 


40 


3626. 


8 


1051 


4 


6128-fi 


40 


4370 


1 


1476 


4 


6949 


5 


40 


5219- 


7 


2021 


1 


7717. 


2 


50 


3638. 


5 


1057 


7 


6143. C 


50 

75° 


4383 


3 


1484 


4 


6962 


8 


50 

85° 


5234- 


9 


2031 


4 


7729. 


5 ^ 


65° 


3650 


2 


1063 


9 


6157.] 


4396 


5 


1492 


4 


6976 





5250 


3 


2041 


7 


7741. 


8 ^ 


10 


3661 


9 


1070 


2 


6171-: 


10 


4409 


8 


1500 


5 


6989 


2 


10 


5265 


6 


2052 


1 


7754 


1 ^ 


20 


3673 


7 


1076 


6 


6185-2 


) 20 


4423 


1 


1508 


6 


7002 


4 


20 


5281 





2062 


5 


7766 


3 i 


30 


3685 


4 


1082 


9 


6199-2 


I 30 


4436 


4 


1516 


7 


7015 


6 


30 


5296 


4 


2073 


0,7778 


6 h 


40 


3697 


2 


1089 


3 


6213-2 


^ 40 


4449 


7 


1524 


9 


7028 


8 


40 


5311 


9 


2083 


57790 


8 ' 


50 


3709 





1095 


7 


6227-2 


I 50 
I 76° 


4463 


.1 


1533 


1 


7041 


9 


50 

86° 


5327 


4 


2094 


1 


7803 


^ 


66° 


3720 


9 


1102 


2 


6241-2 


4476 


1541 


.4 


7055 


C 


5343 





2104 


7 


7815 


2 , 


10 


3732 


7 


1108 


6 


6255-2 


2 10 


4489 


-9 


1549 


•7 


7068 


2 


10 


5358 


6 


2115 


3 


7827 


4 J 


20 


3744 


6 


1115 


1 


6269-] 


L 20 


4503 


■ 4 


1558 





7081 


3 


20 


5374 


2 


2126 


C 


7839 


6 


30 


3756 


5 


1121 


7 


6283-] 


L 30 


4516 


9 


1566 


-3 


7094 


4 


30 


5389 


9 


2136 


7 


7851 


7 


40 


3768 


5 


1128 


2 


6297-{ 


} 40 


4530 


■ 4 


1574 


• 7 


7107 


■ 5 


40 


5405 


-6 


2147 


.5 


7863 


8; 


50 


3780 


4 


1134 


-8 


6310 -{ 


) 50 

3 77° 


4544 


.0 


1583 


1 


7120 


.5 


50 

87° 


5421 


-4 


2158 


-4 
-2 


7876 





67 


3792 


4 


1141 


-4 


6324.i 


4557 


6 


1591 


.6 


7133 


.6 


5437 


.2 


2169 


7888 


•li, 


10 


3804 


4 


1148 





6338-' 


1 10 


4571 


.2 


1600 


•1 


7146 


6 


10 


5453 


-1 


2180 


-2 


7900 


•1 


20 


3816 


4 


1154 


7 


6352-( 


3 20 


4584 


8 


1608 


.6 


7159 


6 


20 


5469 


-0 


2191 


-1 


7912 


2 


30 


3828 


4 


1161 


3 


6366-^ 


I 30 


4598 


.5 


1617 


•1 


7172 


6 


30 


5484 


-9 


2202 


-2 


7924 


.3 


40 


3840 


5 


1168 


1 


6380-1 


J 40 


4612 


.2 


1625 


.7 


7185 


.6 


40 


5500 


9 


2213 


-2 


7936 


3 


50 


3852 


6 


1174 


8 


6394-: 


L 50 

3 78° 


4626 


.0 


1634 


4 


7198 


6 


50 

88° 


5517 


.0 


2224 


-3 


7948 


3 


68° 


3864 


7 


1181 


.6 


6408 . { 


4639 


-8 


1643 


.0 


7211 


■ 6 


5533 


1 


2235 


-5 


7960 


•3? 


10 


3876 


8 


1188 


• 4 


6421. J 


5 10 


4653 


-6 


1651 


.7 


7224 


.5 


10 


5549 


.2 


2246 


-7 


7972 


3' 


20 


3889 





1195 


.2 


6435. ( 


3 20 


4667 


-4 


1660 


.5 


7237 


• 4 


20 


5565 


4 


2258 


-0 


7984 


.2 


30 


3901 


2 


1202 


.0 


6449.^ 


l 30 


4681 


-3 


1669 


.2 


7250 


4 


30 


5581 


.6 


2269 


-3 


7996 


•2t 


40 


3913 


•4 


1208 


• 9 


6463-: 


L 40 


4695 


-2 


1678 


.1 


7263 


• 3 


40 


5597 


8 


2280 


6 


8008 


•1 


50 


3925 


.6 


1215 


8 


6476-1 


3 50 
3 79° 


4709 


-2 


1686 


9 


7276 


•1 


50 

89° 


5614 


-2 


2292 





8020 


^ 


69° 


3937 


.9 


1222 


.7 


6490. ( 


4723 


-2 


1695 


.8 


7289 


.0 


5630 


-5 


2303 


• 5 


8031 


9 


10 


3950 


.2 


1229 


.7 


6504-^ 


I 10 


4737 


.2 


1704 


7 


7301 


-9 


10 


5646 


-9 


2315 


-0 


8043 


8^ 


20 


3962 


.5 


1236 


7 


6518-; 


L 20 


4751 


.2 


1713 


•7 


7314 


-7 


20 


5663 


-4 


2326 


-6 


8055 


7i 


30 


3974 


• 8 


1243 


.7 


6531-J 


3 30 


4765 


.3 


1722 


•7 


7327 


-5 


30- 


5679 


-9 


2338 


.2 


8067 


5f 


40 


3987 


.2 


1250 


-8 


6545-! 


5 40 


4779 


.4 


1731 


.7 


7340 


-3 


40 


5696 


•4 


2349 


.8 


8079 


3- 


50 


3999 


• 5 


1257 


-9 


6559-: 


L 50 

3 80° 


4793 


.6 


1740 


8 


7353 


-1 


50 
90° 


5713 


-0 


2361 


.5 


8091 


2 


70° 


4011 


.9 


1265 


-0 


6572.} 


4808 


•7 


1749 


-9 


7365 


9 


5729 


-7 


2373 


3 


8103 


.0 


10 


4024 


4 


1272 


.1 


6586.^ 


t 10 


4822 


-0 


1759 


-0 


7378 


7 


10 


5746 


3 


2385 


• 1 


8114 


7 


20 


4036 


8 


1279 


■ 3 


6600.] 


L 20 


4836 


-2 


1768 


-2 


7391 


4 


20 


5763 


1 


2397 





8126 


5.i 


30 


4049 


• 3 


1286 


-5 


6613.' 


1 30 


4850 


-5 


1777 


.4 


7404 


1 


30 


5779 


9 


2408 


9 


8138 


2 i 


40 


4061 


• 8 


1293 


-7 


6627.: 


I 40 


4864 


.8 


1786 


7 


7416 


8 


40 


5796 


7 


2420 


9 


8150 


0) 


50 


4074 


• 4 


1300 


9 


6640.1 


) 50 
i 81° 


4879 


-2 


1796 





7429 


5 


50 


5813 


6 


2432 


9 


83 61 


7 


711 


4086 


.9 


1308 


-2 


6654.^ 


4893 


6 


1805 


3 


7442 


2 


91° 


5830 


5 


2444 


9 


8173.4 , 



518 



TABLE III.— SWITCH LEADS AND DISTANCES. 
LEAD-RAILS CIRCULAR THROUGHOUT; GAUGE 4' S¥'. See § 262- 



'^o' 

l(«). 


Frog Angle 


Lead (/.)! ^''^'^ 


Radius of 
Lead-rails 
(r,Eq.78). 


Log r. 


Degree of 
Curve (d). 


Irog 
No. 

(n). 


! 4 


14° 15' 00' 


37-67 


37 


• 38 


150-67 


2.17801 


38° 46' 


4 


4.5 


12 40 59 


42-37 i 


42 


• 12 


190. 69 




28032 


30 24 


4.5 


5 


11 25 16 


47. 08 


46 


• 85 


235-42 




37183 


24 32 


5 


5.5 


10 23 20 


51.79 


51 


-58 


284-85 




45462 


20 13 


5.5 


6 


9 31 38 


56-50 


56 


-30 


339-00 




53020 


16 58 


6 


6.5 


8 47 51 


61-21 


61 


-03 


39785 




59972 


14 26 


6.5 


7 


8 10 16 


65-92 


65 


.75 


461 • 42 




66409 


12 26 


7 


i 7.5 


7 37 41 


70-62 


70 


-47 


529^69 




72402 


10 50 


7.5 


8 


7 09 10 


75-33 


75 


19 


602 • 67 




78007 


9 31 


8 


85 


6 43 59 


80-04 


79 


-90 


680^36 




83273 


8 26 


8.5 


9 


6 21 35 


84-75 


84 


• 62 


762^75 




88238 


7 31 


9 


, 9.5 


6 01 32 


89-46 


89 


• 33 


849 85 




92934 


6 45 


95 


10 


5 43 29 


94.17 


94 


• 05 


941 • 67 


2 


97389 


6 05 


10 


ao.5 


5 27 09 


98-87 


98 


-76 


1038. 19 


3 


01627 


5 32 


10.5 


111 


5 12 18 


103-58 


103 


-47 


1139-42 




05668 


5 02 


11 


111. 5 


4 58 45 


108-29 


108 


■ 19 


1245-36 




09529 


4 36 


11.5 


,12 


4 46 19 


113. 00 


112 


-90 


1356-00 


3 


13226 


4 14 


12 


TUR> 


rOUTS WITH STRAIGHT 


POINT-RAILS AND STRAIGHT FROG- 


RAILS: GAUGE 


4' 8i". See § 265- 


'Frog 

No. 
in). 


Switch 

Point 

Angle 

(a). 


L'gth 

of 
Switch 
Point 

(DN). 


L'gth 

of 
Str'g't 
Frog- 
raiU/). 


Lead 

(L) 

(Eq. 

90). 


Chord 
(ST) 
(Eq. 
88). 


Radius of 
Lead- 
rails 
(r, Eq. 
87). 


Log r. 


Degree 

of 
Curve 


Frog 
No. 
(n). 


\ 4 


3° 40' 


7.5 


1-50 


32-20 


23-09 


125-21 


2.09764 


47° 05' 


4 


< 4-5 


3 40 


7.5 


1 


69 


34 


• 29 


25 • 


03 


159- 


25 




20208 


36 36 


4.5 


. 5 


2 45 


10-0 


1 


87 


41 


• 85 


29- 


88 


• 197- 


65 




29589 


29 22 


5 


• 5.5 


2 45 


10.0 


2 


06 


44 


• 16 


32- 


03 


240- 


44 




38100 


24 00 


5.5 


6 


1 50 


15.0 


2 


25 


56 


-00 


38- 


66 


288- 


09 




45953 


19 59 


6 


' 65 


1 50 


15.0 


2 


44 


58 


-84 


41- 


34 


340- 


19 




53172 


16 54 


6.5 


, 7 


1 50 


15.0 


2 


62 


61 


-65 


43- 


98 


397- 


65 




59950 


14 27 


7 


' 7.5 


1 50 


15-0 


2 


81 


64 


-38 


46- 


50 


460- 


00 




66276 


12 29 


7.5 


8 


1 50 


15.0 


3 


00 


67 


-04 


48- 


99 


527- 


91 




72256 


10 52 


8 


8-5 


1 50 


15-0 


3 


19 


69 


• 60 


51- 


38 


600- 


94 




77883 


9 33 


85 


9 


1 50 


15-0 


3 


37 


72 


20 


53- 


80 


681- 


16 




83325 


8 25 


9 


' 9.5 


1 50 


15-0 


3 


56 


74 


• 70 


56- 


11 


767 ■ 


11 




88486 


7 28 


9.5 


10 


1 50 


15. 


3 


75 


77 


• 04 


58- 


28 


858 ■ 


14 




93356 


6 41 


10 


10.5 


1 50 


150 


3 


94 


79 


• 51 


60- 


57 


959- 


00 


2 


98182 


5 59 


10.5 


ill 


1 50 


15.0 


4 


12 


81 


-82 


62- 


69 


1065- 


52 


3 


02756 


5 23 


11 


11.5 


1 50 


15.0 


4 


• 31 


84 


-09 


64- 


78 


1180. 


16 


3 


07194 


4 51 


11.5 


•12 


1 50 


15. 


4 


-50 


86 


• 16 


66- 


67 


1299. 


93 


3 


11392 


4 24 


12 


! 


TRIGONOMETRICAL FUNC^ 


nONS OF THE FROG ANGLES (F). 


Frog 

No 


Frog Angle 


Nat. 


Nat. 


Log 


Log 


Log 


Log 


Frog 

No 


(n). 


(F). 


sini^. 


cos F. 


sin F. 


cos F. 


cot F. 


vers F. 


(n).* 


4 


14° 15' 00" 


24615 


•96923 


9-39120 


9.98642 


10.59522 


8 •48811 


4 


4.5 


12 40 49 ' 


21951 


•97561 


•34145 




98927 




6478ki 




38721 


4.5 


5 


11 25 16 i 


19802 


•98020 


•29670 




99131 




69461 




29670 


5 


5.5 


10 23 20 


•18033 


•98360 


•25606 




99282 




73675 




21467 


5.5 


6 


9 31 38 i 


•16552 


•98621 


•21884 




99397 




77513 




13966 


6 


65 


8 47 51 ' 


-15294 


98823 


•18453 : 


99486 




81033 




Q7058 


6.5 


7 


8 10 16 


•14213 1 


-98985 


•15268 


99557 




84288 


8 


00655 


7 


7.5 


7 37 41 


•13274 


-99115 


•12301 




99614 




87313 


7 


94691 


7.5 


' 8 


7 09 10 


12452 


•99222 


•09522 




99660 




•90138 




89110 


8 


i 85 


6 43 59 


-11724 1 


•99310 


•06909 




99699 




.92790 




83864 


85 


' 9 


6 21 35 


-11077! 


.99385 


•04442 




99732 




•95289 




78915 


9 


1 9.5 


6 01 32 i 


• 10497 ; 


•99448 


9^02107 




99759 




•97652 




74232 


9.5 


110 


5 43 29 


•09975 


•99501 


8-99891 




99783 


10 


-99892 




69788 


10 


10.5 


5 27 09 ' 


• 09502 ! 


■99548 


-97781 




99803 


11 


-02021 




65560 


! 10.5 


11 


5 12 18 


•09072 


•99588 


-95770 




•99820 




■04050 




•61528 


11 


ai.5 


4 58 45 


-08679 


.99623 


-93848 




.P9836 




•05987 




•57676 


11.5 


12 


4 46 19 


•08319 


•99653 


8-92007 


9 


•99849 


11 


•07842 


7 


•53986 


1 12 


519 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 



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I-H 


f-l 

o 


o 

ID 


ID Oolo 
CO i-l|C 


^0. 

cq 


CO 


CO 

I-H 


CO ID 

-i^ll i-l 




O 


I-H 


o 


o oic 


O 


o 


1— 1 


r-H CM 




ID 


tD 


o 


o 

c 


o 
o 


ID CM 


o 

CO 


o 


ID CM 
I-H ID 


^ 


"cD 


00 
CO 


00 
(N 


ID 


o 
o 


00 Oi 

^•-i CO 


o 


00 


CD CD 
ID CM 




o 
O 


o 


O 


c 


o 


o 


o 


I-H 


f— 1 


I-H <M 



O 
CO 


CO 


ID 

1— 1 


O 

o 


"c>. 

CM 


o 

CM 


1— 1 


o 
o 


o 
O 


O 


C 


c 



ID l> O CN ID 

i-t O CO CsJ '^ 



rH CO l>- '^ CO 
ID I— I CO O CO 



O i-l I-H CM CN 



w 



"t^ c 

O CO 


o 

c 


"co r> 

rH O 


c 
c 


l°oo 


c 



O- 



ID cq 

rH CM 



I-H ^ 
I-H CN 



t>- ID CM O t-> 
O •<# ID CO CO 



00 00 i-l t>. ID 

ID I-H ^ O CO 



COOOOr-HrHCNCq 



IT 


o 
o 


o 
1*^ 


CM 
ID 


ID 


o 


o 
o 


CM 
CM 


ID 

I-H 


co 


o 

CO 


CC 


o 


'r> 


CD 
i-H 


00 
CM 


CO 


o 
o 


05 

I-H 


I-H 


ID 

o 


CM 
CO 



OOOOr-Hi-Hi-HCNCN 



c 


ID 


<N 
CM 


o 

CO 


o 


ID 

r-i 


ID 


o 
o 


eo 


ID 


o 

CM 


c 


CO 


05 




00 
CN 


f-l 


CD 
ID 


ID 

I-H 


ID 
CO 


00 
ID 


•^1 
CM 


°o 


o 


O 


o 


O 


O 


o 


I-H 


t-i 


I-H 


Csl 



C^r^wwTt<i0CDi>.Qca5O 



520 



to 

o 



TABLE IV.— ELEMENTS OF TRANSTTION CURVF-S. 



00505t>-«^COOqC^C»05 
rH i-H rH I— I Cq CM 



KMICOICSJICOIt* 00 i-HIOJ CDlrH 

I CD CO eg CO CO ir^ eg lo o C 
CO CO CO rH o. 03 00 ■^ c» eg 

I C>- '^Jl CO eg 'gH CO CO O r-H CO 

I 00 05 as o o o o i-H I— 1 1— 1 



locococooococ^egco 
mo.cDcoc330cgoo>— I 
oegc^coc35C33coi— itnoa 

I— I eg ■^ £>■ rHin o 

rHrHeg 



I C35OC»CD0-jC0C0^i~II>- 

030200505 0^05050300 

I 05 C5 05 03 05 03 05 05 03 05 

I 05 05 05 05 05 05 05 05 05 05 



li-iicg eg 00 o coioi coicoio 
ooooscocO'^cocoego 

0500.C351C51— IIO-^OCO 
COi— lr—iCOr-IC00003C5i>- 

coi— i'^cD00O5Oi— legco 

t^ OO 00 00 00 00 05 05 05 05 



O5icoioice COIlO C^ COICO 
O o>0505C>ioegt>-Or-H 

M O5O5O5O5C3O50000!>- 
Q 05 05 05 05 05 05 05 05 05 



ICO r-i eg CD tJi micoi^ r-i t> 
•<^co«5comi— irHcoint^ 
Oi—ieq-siicooscNjincsco 
ooooooi— ii— ii-neg 



^^ 1 lO "vH CO:CO CO -^ OOl.-llCOI'^Jl 

CO 1 coc»cx5tococ>-cococoeg 

}r: 1 OOCNl'^COOC^lCvlOCOC^ 

^ 1 [>. CO CO c^ CO e<j c>- 05 CO U5 

1 05 O in C35 CO CO 00 O CN "^iH 


O 1 -^ u:> CD CD t^ t- t>. 00 00 00 


■©■ 1 ic5;cD loico t^it- m egit- 1- 

CO 1 05 05 CO to O r-H t^ CO ITS CO 



<o 




bo 




cz 






mmooioinoomio 


^«-e- 


.-l'<jlCOCO'«:*<r-HOOrH'>:J< 


^i^ 


oOrHCqcomt^osr-ico 


<v 


i-HrH 


O 




c 


r-HegcO':*<ir5CDt>-ooo50 


o 


l-t 


Q- 







O- 


• 


3 


si 




;-i. 


o 






A 

OD 


CI 


4A 


o 


V 


a 


£ 


c 




X 




-^j 


« 


■§ 


^ 


-^J 


<u 


G 


A 




o 


g 


H 


«? 



QC 



O 



«3 



W 



W 



O 



eg 
eg 


in 

o 




eg 

CO 


1— 1 


o 
o 


in 


o 

CO 


in 


OjO 


"^CD 


in 
eg 


CO 


o 


eg 


o 

00 


CO 
CO 


eg 

CO 


CO 

eg 


s§ 


00 


00 


t^ 


i> 


CO 


ITS 


^ 


CO 


eg 


Jo 


CO 




eg 

o 


in 

I— 1 


o 

CO 


to 


o 

o 


in 

t—i 


CO 


o 
o 


o 
o 


I-l 


00 


lO 

r-H 


CD 
CO 


eg 
in 


CO 

o 


o- 

r-t 


r-H 




o 
o 


in 

r— < 



o-cDoin^riicoeg i-h"o 



c^ o in o in 

"fii CO r-t (Z> ■^ 



CO eg rH in CO 
*;{« eg in I— I CO 



o in 

CO i-l 



t>. CD 

^ in 



inin^Tjieocgi— ii-H 



oao 
oaco 






o in o 

O r-H CO 



o CO t^ 
CO o CO 



in o 

^ o 



CO m 

o eg 



^ ^ CO CO eg 



in 

r-H 


o 

eo 


o 
o 


i-l 


s 


o 

o 


1— 1 


o! 


o 



o in 

o ■<* 



O CO 

o o 



"in 

r-H 


o 
o 


in 


o 

CO 


in 
I-l 


o 
o 


§ 


o 

CO 




O 

o 


in 

I— 1 


eg 


o 
o 


CO 
CO 


CSI 

o 


CO 


s 


o 
o 


eg 
in 


00 


o 
in 


CO 

in 


CO 


CO 


eg 


eg 


r-l 


J> 


o 


o 


i-H 


C<! 


CO 



O in o 
CO •«* O 



eg CO o 
eg o •«* 



in o 

I— I CO 



i-< CO 



c>g eg 1-4 rH o 



ojo 

ojo 



olin 



in o in 

^ CO r-H 



CO c^ o 
CO eg eg 



o'o rH eg CO ■«;>< 



in o in o 

^ CO I— I o 



CO C^ CO o 

CO I— I in CO 



1-4 o o 



o in 

CO ^ 



t>. 00 
CO 1-4 



o in o 

O r-4 CO 



in CO eg 
o in m 



OJO rH eg eg CO -^ 



O in o'o 

rH CO O 

"in rH eg!o 
in Ti4 eg o 

o i 

o o o o 



o 
o 


in o 

'^ CO 


in 

rH 


o 
o 


in 


00 
C<1 


o 

CO 


CO eg 
o -^ 


CO 

eg 


in 

r-i 


00 

o 


o 



O rH rH eg CO ^ in 



in o 

rH O 



CO in 
eg n^, 



O CO '!i4 



eg CO 

eg '!i< 



o in 

O r-i 



O CO 

eg in 



o in CO 
00 •'dH in 



C^ CO '34 

CO eg rH 



OOrHrHCNCO'^in 



o in o 

O ■'^ CO 



in CO E>» 
rH CO in 



in o 

rH O 



CO O 

eg o 



in o CO 

•«i4 CO rH 



00 eg rH 
CO eg rH 



ooorHegegco^in 



o o 



in o in 

•«:f4 O rH 



o in 

CO Ti4 



O CO CO 

o rH eg 



O rH t^ 
CO rH in 



O OO O rH rH eg CO CO t:H 



•II 



O^HWCO^JCCOt^GC 



521 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 



^ 


1 

O CO !>. CO t>- 0> O CM CO 'd' 

oa)i>..-icoina2(Mt>.co 
ooaosost^^cocncoo 

CS]r}<t>.05CM^D-OCSi^ 
f— 1 1— 1 1— 1 r-l CM <N 


5^ 

ho 

o 


•■^lO OOIOOICM 03ID^ I> D^ CO 
I>.t>.COCOCDt>-OCOOOrH 
t>-COCO^t>,ir5C<]^£>-CO 
COeOOOrHt>.0300^00rH 
Ot^i— llOt«-05.— ICO^CO 




a>050000f-ii— ii— ii— 1 

i 


5^ 


O5ir)t>.,-icqcoooo5cqt>. 

O';J<cqt*-CDOOCi5i0i— 1 

,-liniO<N03C»CMOI>-CO 


OOt-HCOlOOSlOCSlOi-l 

i-Hcqco-«^ 


-e- 

t- 
> 

ho 

3 


CO 00 t>- osicq 05 <Nia3 oico 
cocooococsiooooaatno 
o^cDOC<ieocMa3r-tco 
ooeocooocoCN]c>.cocoio 
ir>iDrHino3Csi'sj<cDooo 


ir5COt^t^t».OOOOOOC003 


-e- 

o 
o 

ho 
o 

—I 


100 lOIOI""* I>.|CDIOIOIr-i CO 
C3500'«;t<COCSJCDO5Cs|CtiO5 
CX)O3O500COCNJCOe0iOO. 
0503505030303 CO t^CO'^lH 
05 05 05 05 C3 05 05 05 05 05 


05 05 05 OO 05 05 05 05 05 05 


t 

ho 


'«d< CSI 0!05 OICOIC- 00 ^\G 
COCSCOOqt^COCDCSSCO'^ 

oc>.ooo!ooeoo5Csj'stH 

's;< r— 1 .-- 1 T:f r— 1 CO 00 00 CO CD 
05'«^t>C3r-HCqCO'^lDCO 


o 


t>C00000C35O3O5O5O3C35 


-e- 

CO 

o 
o 


IC3ICOICO CMIrJllCN COlO 05 O 
050500CO.-iCOO.-ICOt«- 
030505CD0500I>-lOCvlCO 
05 05 03 05 05 03 05 05 05 00 


C3 




-0- 


lt>. CQICOIr-lllOKM 05 O C^lt"- 

oocD<Mi>-oc<Ji-icr)CqrH 

OC<IUDOOCOCO':^OCOCO 
OOOOr-lr-HCqCOCOTj< 






Total 
Central Angle 

<t> 


oooooooooo 
eocooococooococo 

o 
OrHOOir5i>.o^oooqc^ 

1— 1 1— 1 rH CM CM 


"o 


i-icaco'«#iocDt>.ooo50 



I 
o 







in 


CSI 

Til 


o 

o 


o 


CM 


CD 

o 


o 

CO 


o 
o 


o 

CO 


il 


§ 




o 


COOOOiOCNOC>-lOCSIO 
lOlOCOr-l-<iHOOOir>CO 


§ 






t-COiO"«#CSIi-H05C>-TJ<CMio 




ocQir)05ir5oooc 
coiDi-ieoocoococ 


c 
o 


o 

o 




o: 


ir5C--oc<jioi>.ocMm 

COCOeOi-H-^OCSlCSlr-l 


o 

o 


o 

CO 






""^COCqrHOSOOCD^CSJ 

rH 1— t r-H 1— 1 


o 


cq 




"loocorHoooo olo o 

'«#i-lOOOCOOCOO 0|0 CO 


C 


00 


t-incsiob-iocNio 

OO'^^COOCOlOO 


o 

o 


m c^ 

i-l 00 




f-i0030oc-iococq 

r-H i-H 


o 


cq -<* 




V r ■ 


c 




IT) Til CSI O O O O 
O CO O CO O CO c 


o o o o 

O O CO o 




t>» 


O CQ in t^ O CM IT! 

o ph i-i o lo cq ^ 


O O t- ID 
O O O CM 


-^i 

C 
C 

sx 

^ 




03 00 b- CD -"^ CO i-H 


O CM rt< CD 




O rH O O O C 

CO o CO o CO c:. 


c 


O O O O 
O CO O CO 


^ 


o 


oq o E>- lo C5 c 
"^ CD a G in en 




in !>. o cq 

■<^ CO T*< in 






o 
CD CO IT) ^ CM I—; 


o 


rH 00 in t> 


o 




^ • p ■ 


c 




o o o o c 

O CO O CO o 


s 


iO O O O -i* 

o CO o CO in 


'c 


IC 


*ic5 t- o CM m 

rji O CN CM r-l 


o 

o 


o t^ in cq 03 
CO o in in in 






*^ •«;*( CO <M i-H 


o 


rH CO "^ CD 00 




o o o o 

CO O CO o 


o 

c 


o o o o in 00 

O CO O CO in r-H 


03 


Tt* 


t- lO CM O 

o CO ir> o 
o 

CO Cq i-H 1— 1 


o 


in i>. o CM ^ t> 

I-H CO rH in ^ ■«# 
r-l CM Tj< in b- 05 




o o o 

O CO o 


o 

o 


O O O O CO r-H o 

o CO o CO in CM Tj< 




CO 


o CM ir> 
in C<1 rj< 


§ 


o c- in CSI C35 t- ■«# 

O O Cq in CSJ 1— 1 rH 


a 




rH 1— 1 O 


o 


I-H CSI CO ^ CD OO O 

9 I-H 




o o o 
CO c o 


ooooooT*<ino 
ocoocomc^^o 


.2 


w 


Csl o o 
in coKcD 


inc>^ocq^i>.05cq 
■^co-'diini-H^CMcq 






o olo 


Oi-HCMCOIOCOOOO 

i-H 




c 

c 


i 


oooooscoooot^ 
oeoocoincqinOi-H 




iH 


lO 


c 
o 


Ot>inCM03t>-T;J<CSI05 

cooinininr-H^csio 






o 
o 


o 


Or-lrHCSICOlOCOOOO 

» 1—1 




o 

o 


ooooooocoinom 
ocoocoocMini-Hcoco 




a 


c 

o 


ini>.o<Mmt>-o5CM^cD 
f-icot-iin-^Ttfincoinco 


1 


o 
o 


OOi— li-IC^1CO^COt-CT> 


1 


•i 

c3 


a 


iH 


N 


CO 


Tt< 


»0 


?0 


r^ 


00 


Oi 


O 



522 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


100 


00 000 


043 


087 


130 


173 


216 


260 


303 


346 


389 










1 














, 43 43 42 41 


101 


432 


475 


518 561 


604 


646 


689 


732 775 


817 


.1 


4.3 


4.3 


4.2 


4.1 


102 


860 


902 


945 


987 


*030 


*072 


*114 


*157*199 


*241 


• 2 


8 


7 


8 


6 


8 


4 


8 


.2 


103 


01 283 


326 


368 


410 


452 


494 


536 


5781 619 


661 


• 3 


13 





12 


9 


12 


6 


12 


3 


104 


703 


745 


787 828 


870 


911 


953 


994i*036 


*077 


• 4 


17 


4 


17 


2 


16 


8 


16 


.4 


105 


02 119 


160 


201| 242 


284 


325 


366 


4071 448 


489 


• 5 


21 


7 


21 


5 


21 


.0 


20 


.5 


106 


530 


571 


612 653 


694 


735 


775 


8161 857 


898 


6 


26 


1 


25 


8 


25 


.2 


24 


.6 


107 


938 


979*019*060 


*100 


*141 


*181 


*22ll*2e2 


*302 


•7 


30 


4 


30 


.] 


29 


.4 


28 


• 7 


108 


03 342 


382 422 


463 


503 


543 


583 


623; 663 


703 


8 


34 


8 


34 


•4 


33 


.6 


32 


8 


109 


742 


782 822 


862 


901 


941 


981 
375 


*020|*060 
415 454 


*100 
493 


.9 


39 


1 


38 


■7 


37 


8 


36 


9 


110 


04 139 


178! 218 


257 


297 


336 


40 40 39 38 
















1 




111 


532 


57l 610 


649 


688 


727 


766 


805! 844 


883 


.1 


4.0 


4.0 


8.9 


3. 8 


112 


922 


960 999 


*038 


*076 


*115 


*154 


*192 *231 


*269 


• 2 


8 




8 


.0 


7 


8 


7.6 


113 


05 308 


346 384 


423 


461 


499 


538 


576 614 


652 


• 3 


12 




12 


.0 


11 


• 7 


11 .4 


114 


690 


728 766 


804 


842 


880 


918 


956 994 


*032 


• 4 


16 


2 


16 





15 


-6 


15.2 


115 


06 070 


107 145 


183 


220 


258 


296 


333 371 


408 


• 5 


20 


2 


20 





19 


5 


19.0 


116 


446 


483 520 


558 


595 


632 


670 


707 1 744 


781 


.6 


24 


. 3 


24 


.0 


23 


.4 


22.8 


117 


818 


855| 893 


930 


967 


*004 


*040 


*077*114 


*151 


.7 


28 


• 3 


28 


.0 


27 


3 


26.6 


118 


07 188 


225 


261 


298 


335 


372 


408 


445 481 


518 


8 


32 


4 


32 





31 


• 2 


30.4 


119 


554 


591 


627 


664 


700 


737 


773 


809 845 


882 


• 9 


36 


•4 


36 


.0 


35 


• 1 


34.2 


130 


918 


954 


990 


*026 


*062 


*098 


*134 


*170 


1*206 


*242 


_ 


121 


08 278 


314! 350 


386 


422 


457 


493 


529 


564 


600 


• 1 


3.7 


«5/ 

3.7 


36 


«50 

3.5 


122 


636 


6711 707 


742 


778 


813 


849 


884 


! 92C 


955 


.2 


7 


5 


7 


4 


7 


2 


7.0 


123 


990 


*026*061 


*096 


*131 


*166 


*202 


*237,*272 


*307 


.3 


11 


2 


11 


1 


10 


8 


10.5 


124 


09 342 


3771 412 


447 


482 


517 


552 


586 


621 


656 


• 4 


15 





14 


8 


14 


4 


14.0 


125 


691 


725i 760 


795 


830 


864 


899 


933 


968 


*002 


• 5 


18 


7 


18 


5 


18 





17.5 


126 


10 037 


07l| 106 


140 


174 


209 


243 


277 


312 


346 


• 6 


22 


5 


22 


2 


21 


6 


21.0 


127 


380 


414| 448 


483 


517 


551 


585 


619 


653 


687 


• 7 


26 


2 


25 


9 


25 


2 


24.5 


128 


721 


755 


789 


822 


856 


890 


924 


958 


991 


*025 


8 


30 





29 


6 


28 


8 


28.0 


129 


11 059 


092 


126 


160 


193 


227 


260 


294 


327 


361 


9 


33 


7 


33 


3 


32 


4 


31.5 


130 


394 


427 


461 


494 


528 


561 


594 


627 


661 


694 


35 34 33 33 








■ 
















131 


727 


760 793 


826 


859 


892 


925 


958 


991 


*024 


• 1 


3 4 


3 4 


33 


3.2 


132 


12 057 


090i 123 


156 


189 


221 


254 


287 


320 


352 


• 2 


6 


9 


6 


8 


6 


6 


6 


4 


133 


385 


418 450 


483 


515 


548 


580! 613 


645 


678 


.3 


10 


3 


10 


2 


9 


9 


9 


6 


134 


710 


743 775 


807 


840 


872 


904; 937 


969 


*001 


•4 


13 


8 


13 


6 


13 


2 


12 


8 


135 


13 033 


065, 097 


130 


162 


194 


226! 258 


290 


322 


.5 


17 


2 


17 





16 


5 


16 





136 


354 


386i 417 


449 


481 


513 


545' 577 


608 


640 


• 6 


20 


7 


20 


4 


19 


8 


19 


2 


137 


672 


703 735 


767 


798 


830 


862' 893! 925 


956 


.7 


24 


1 


23 


8 


23 


1 


22 


4 


138 


988 


*019 *05li*082 


*11R 


*145 


*176 *207*239 


*270 


8 


27. 


6 


27 


2 


26 


4 


25 


6 


139 


14 30l 


332 


364 


395 


426 


457 


488 519 550 


582 


• 9 


31. 


6 


30 


6 


29. 


7 


28 


8 


140 


613 


644 


675 


706 


736 


767 


798i 829 860 


891 


3T 31 30 39 






1 








■ 




141 


922 


955 9831*014 


*045 


*075 


*106 *137*167 


*198 


• 1 


3.1 


3 1 


3-0 


2.9 


142 


15 229 


259 2901 320 


351 


381 


412 442 473 


503 


.2 


63 


6 


2 


6. 





58 


143 


533 


564 594 624 


655 


685 


715 745 776 


806 


• 3 


9.4 


9 


3 


9. 





8.7 


144 


836 


866; 896 926 


956 


987 


*017 *047 *077 


*107 


■ 4 


12.6 


12 


4 


12 





11.6 


145 


16 137 


166i 196! 226 


256 


286 


316 346 376 


405 


.5 


15.7 


15 


5 


15. 





14.5 


146 


435 


465' 494 


524 


554 


584 


613 643 672 


702 


■ 6 


18. 9 


18 


6 


18. 





17.4 


147 


731 


761i 791 


820 


849 


879 


908 938 967 


997 


•7 


22.0 


21 


7 


21 





20.3 


148 


17 026 


055| 085 


1 114 


143 


172 


202 2311 260 


289 


• 8 


25.2 


24 


8 


24 





23.2 


149 


318 


348i 377 


406 


435 


464 


493 522' 551 


580 


• 9 


28.3 


27 


9 


27. 





26.1 


150 


609 


638 


667 


696 


725 


753 


782! 8ll 


840 


869 




N. 





1 


2 


3 


4 


5 


6 


7 


s 


9 








P. 


P 


• 









523 









TABLE y 


\— LOGARITHMS 


OF NUMBERS 


. 












N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


150 


17 609 


638 


667 


696 


725 


753 


782 


811 


840 


869 


























29 28 27 


151 


897 


926 


955 


984 


*012 


*041 


^070 


*098 


*127 


*156 


.1 


2-9 


2.8 


2.7 


152 


18 184 


213 


241 


270 


298 


327 


355 


384 


412 


440 


.2 


5 


.8 


5.6 


5 


.4 


153 


469 


497 


526 


554 


582 


611 


639 


667 


695 


724 


.3 


8 


.7 


8.4 


8 


1 


154 


752 


780 


808 


836 


864 


893 


921 


949 


977 


*005 


• 4 


11 


.6 


11.2 


10 


8 


155 


19 033 


061 


089 


117 


145 


173 


201 


229 


256 


284 


.5 


14 


= .5 


14.0 


13 


.5 


156 


312 


340 


368 


396 


423 


451 


479 


507 


534 


562 


.6 


17 


• 4 


16-8 


16 


• 2 


]57 


590 


617 


645 


673 


700 


728 


755 


783 


810 


838 


.7 


20 


3 


19-6 


18 


9 


158 


865 


893 


920 


948 


975 


*003 


*030 


*057 


*085 


ni2 


.8 


23 


• 2 


22.4 


21 


.6 


159 


20 139 


1G7 


194 


221 


249 
520 


276 
547 


303 


330 


357 


385 


.9 


)2C 


.1 


25.2 


24 


• 3 


160 


412 


439 


466 


493 


574 


601 


628 


655 


26 26 
























161 


682 


709 


736 


763 


790 


817 


844 


871 


898 


924 


.1 


2.6 


2.6 


162 


951 


978 


*005 


*032 


*058 


*085 


*112 


*139 


*165 


*192 


.2 


5 


.3 


5 


• 2 


163 


21 219 


245 


272 


298 


325 


352 


378 


405 


431 


458 


.3 


7 


.9 


7 


8 


164 


484 


511 


537 


564 


590 


616 


643 


669 


695 


722 


•4 


10 


.6 


10 


• 4 


165 


748 


774 


801 


827 


853 


880 


906 


932 


958 


984 


.5 


13 


.2 


13 


.0 


166 


22 Oil 


037 


063 


089 


115 


141 


167 


193 


219 


245 


.6 


15 


• 9 


15 


.6 


167 


271 


297 


323 


349 


375 


401 


427 


453 


479 


505 


.7 


18 


.5 


18 


.2 


168 


531 


557 


582 


608 


634 


660 


686 


711 


737 


763 


.8 


21 


• 2 


20 


8 


169 


788 


814 


840 


885 


891 


917 


942 


938 


994 


*019 


.9 


23 


• 8 


23.4 


170 


23 045 


070 


098 


121 


147 


172 


198 


223 


249 


274 


25 25 24 
























171 


299 


325 


350 


375 


401 


426 


451 


477 


502 


527 


.1 


2-5 


2.5 


2.4 


172 


553 


578 


603 


628 


653 


679 


704 


729 


754 


779 


.2 


5 


•1 


5.0 


48 


173 


804 


829 


855 


880 


905 


930 


955 


9GC 


^005 


*030 


.3 


7 


• 6 


7.5 


7.2 


174 


24 055 


080 


105 


129 


154 


179 


204 


229 


254 


279 


.4 


10 


.2 


10-0 


9.6 


175 


304 


328 


353 


378 


403 


427 


452 


477 


502 


526 


• 5 


12 


•7 


12.5 


12.0 


176 


551 


576 


600 


625 


650 


674 


699 


723 


748 


773 


• 6 


15 


• 3 


15.0 


14.4 


177 


797 


822 


846 


871 


895 


920 


944 


968 


993 


*017 


.7 


17 


.8 


17.5 


16.8 


178 


25 042 


066 


091 


115 


139 


164 


188 


212 


237 


261 


.8 


20 


.4 


20.0 


19.2 


179 


285 


309 


334 


358 


382 


406 


430 


455 


479 
720 


503 
744 


.9 


22-9 


22.5 


21.6 


180 


527 


551 


575 


598 


623 


647 


672 


696 




181 


768 


792 


816 


840 


863 


887 


911 


935 


959 


983 


.1 


2.3 


2.3 


182 


26 007 


031 


055 


078 


102 


126 


150 


174 


197 


221 


.2 


4 7 


4 


6 


183 


245 


269 


292 


316 


340 


363 


387 


411 


434 


458 


.3 


7.0 


6 


9 


184 


482 


505 


529 


552 


576 


599 


623 


646 


670 


693 


.4 


9.4 


9 


2 


185 


717 


740 


764 


787 


811 


834 


858 


881 


904 


928 


.5 


11.7 


11 


5 


186 


951 


974 


998 


^021 


*044 


*068 


=^091 


=^114 


*137 


*161 


.6 


14.1 


13 


8 


187 


27 184 


207 


230 


254 


277 


300 


323 


346 


369 


392 


.7 


16.4 


16 


1 


188 


416 


4.39 


462 


485 


508 


531 


554 


577 


600 


623 


.8 


18. 8 


18 


4 


189 


646 
875 


669 
898 


692 
921 


715 
944 


738 
966 


761 


784 


806 


829 


852 


.9 


21.1 


20-7 


190 


989 


*012 


*035 


*058 


*080 




























191 


28 103 


126 


149 


171 


194 


217 


239 


262 


285 


307 


.1 


2.2 


2.2 


fiJX. 

2-1 


192 


330 


352 


375 


398 


420 


443 


4G5 


488 


510 


533 


.2 


4.5 


4.4 


4. 


3 


193 


555 


578 


600 


623 


645 


668 


690 


713 


735 


758 


.3 


6-7 


6.6 


6. 


4 


194 


780 


802 


825 


847 


869 


892 


914 


936 


959 


981 


•4 


9-0 


88 


8. 


6 


195 


29 003 


025 


048 


070 


092 


114 


137 


159 


181 


203 


.5 


11.2 


11-0, 


10. 


7 


196 


225 


248 


270 


292 


314 


336 


358 


380 


402 


424 


.6 


13.5 


13.2 


12. 


a 


197 


446 


468 


490 


512 


534 


556 


578 


600 


622 


644 


• 7 


15.7 


15.4 


15. 





198 


666 


688 


710 


732 


754 


776 


798 


820 


841 


863 


.8 


18.0 


17.6 


17 . 


t 


199 


885 
30 103 


907 
124 


929 


950 


972 


994 


=^016 


*038 


*059 


*081 
298 

9 


.9 


20.2 


19.8d9.3 


200 


146 


168 


190 
4 


211 
5 


233 
6 


254 

7 


276 
8 




N. 





1 


2 


3 


P. P. 



524 









TABLE v.— LOGARITHMS OF NUMBERS. 


1 ■ -- 
1 
N. 





1 
124 


3 

146 


3 

168 


4 

190 


5 

2ll 


6 

233 


7 
254 


8 

276 


9 


P. P. 


|300 


30 103 


298 


33 31 


: 






















201 


319 


341 


363 


384 


406 


427 


449 


470 


492 


513 


.1 2.2 


2.1 


202 


*35 


556 


578 


599 


621 


642 


664 


685 


707 


728 


.2 4.4 


4. 


2 


203 


749 


771 


792 


813 


835 


856 


878 


899 


920 


941 


• 3 6-6 


6. 


3 


204 


963 


984 


*005 


^^027 


*04S 


*069 


^090 


*112 


*133 


*154 


.4 88 


8. 


4 


205 


31 175 


196 


217 


239 


260 


281 


302 


323 


344 


365 


.5 11.0 


10. 


5 


206 


386 


408 


429 


450 


471 


492 


513 


534 


555 


576 


.6 13-2 


12. 


6 


207 


597 


618 


639 


660 


681 


702 


722 


743 


764 


785 


.715-4 


14. 


7 


•208 


806 


827 


848 


869 


890 


910 


931 


952 


973 


994 


.8 17-6 


16- 


8 


209 
1310 

bii 


32 014 
222 


035 


056 


077 


097 


118 


139 


160 


180 


201 


.9 19-8 


18. 


9 


242 


283 


284 


304 


325 


346 


366 


387 


407 


30 30 






















428 


449 


469 


490 


51C 


531 


551 


572 


592 


613 


.1 2.0 


2.0 


|212 


633 


654 


674 


695 


715 


736 


756 


776 


797 


817 


.2 4.1 


4 





1213 


838 


858 


878 


899 


919 


94C 


960 


980 


'^001 


*021 


.3 6.1 


6 





'1214 


33 041 


061 


C82 


102 


122 


142 


163 


183 


203 


223 


.4 8-2 


8 





.1215 


244 


264 


284 


304 


324 


344 


365 


385 


405 


425 


.5 10.2 


10 


G 


1216 


445 


465 


485 


505 


525 


546 


566 


586 


606 


626 


.6 12.3 


12 





!217 


646 


686 


686 


706 


726 


746 


766 


786 


806 


825 


.7 14.3 


]4 





'218 


845 


865 


885 


905 


925 


945 


965 


985 


*CC4 


*024 


.8 16-4 


■16 





219 
330 


34 044 


064 


084 


104 
301 


123 


143 


163 


183 


203 


222 


.9 18.4 


.18 





242 


262 


28l 


321 


341 


360 


380 


400 


419 


19 19 
























221 


439 


459 


478 


498 


518 


537 


557 


576 


596 


615 


.1 l.c 


19 


222 


635 


655 


674 


694 


713 


733 


752 


772 


791 


811 


• 2 3-5 


3 


8 


, 223 


830 


850 


869 


889 


9C8 


928 


947 


966 


986 


=^CC5 


.3 5.f 


5 


•7 


224 


35 025 


044 


063 


083 


102 


121 


141 


160 


179 


199 


.4 7.J 


I 7 


6 


'225 


218 


237 


257 


276 


295 


314 


334 


353 


372 


S£l 


.5 9.^ 


f 9 


.5 


' 226 


411 


430 


449 


468 


487 


507 


526 


545 


564 


58S 


.6 11.' 


1 11 


■ 4 


227 


602 


621 


641 


660 


679 


698 


717 


736 


755 


774 


.713.( 


3 13 


3 


' 228 


793 


812 


831 


850 


869 


888 


907 


926 


945 


964 


• 8 15. ( 


5 15 


.2 


229 


983 
36 173 


*002 
191 


*021 
210 


*040 
229 


*C59 
248 


^078 


*097 


*116 


*135 


*154 


•9 17. f 


)17 


.1 


1 330 


267 


286 


305 


323 


342 


18^ 18 


i 


















' 231 


361 


380 


399 


417 


436 


455 


474 


492 


511 


530 


.1 i.r 


? 1.8 


232 


549 


567 


586 


605 


623 


642 


661 


679 


698 


717 


.2 3.' 


f 3.6 


233 


735 


754 


773 


791 


810 


828 


847 


866 


884 


903 


.3 5.1 


) 5.4 


234 


921 


940 


958 


977 


996 


*C14 


*C33 


*051 


*07C 


*C88 


.4 7-^ 


I 7-2 


235 


37 107 


125 


143 


162 


180 


199 


217 


236 


254 


273 


.5 9.i 


I 9.0 


236 


291 


309 


328 


346 


364 


383 


401 


420 


438 


456 


.6 11.] 


L 10.8 


237 


475 


493 


511 


530 


548 


566 


584 


ecs 


621 


639 


• 7 12.J 


) 12.6 


238 


657 


678 


694 


712 


730 


749 


767 


785 


803 


821 


.8 14:. i 


i 14.4 


239 

340 

1 


840 
38 021 


858 
039 


876 
057 


894 
075 


912 
093 


930 


948 


967 


985 


^^003 
183 


.9 16. ( 


:I16.2 


111 


129 


147 


165 


17 17 












1 241 


201 


219 


237 


255 


273 


291 


309 


327 


345 


363 


.1 1.^ 


■ 1.7 


242 


381 


399 


417 


435 


453 


471 


489 


507 


525 


543 


.2 3.£ 


) 3.4 


243 


560 


578 


596 


614 


632 


650 


667 


685 


703 


721 


.3 5.1 


> 5.1 


244 


739 


757 


774 


792 


810 


828 


845 


863 


881 


899 


.4 7.C 


6.8 


245 


916 


934 


952 


970 


987 


=^005 


*023 


*C40 


^058 


*076 


• 5 8-7 


8.5 


246 


39 093 


111 


129 


146 


164 


181 


199 


217 


234 


252 


.6 lO-E 


10.2 


247 


269 


287 


305 


322 


340 


357 


375 


302 


410 


427 


.7.12.2 


11.9 


248 


445 


462 


480 


497 


515 


532 


550 


567 


585 


602 


• 8 14. C 


13.6 


249 


620 


637 


655 


672 


689 


707 


724 


742 


759 


776 


.915-7 


15.3 


350 


794 


811 


828 
3 


846 
3 


863 
4 


881 
5 


898 
6 


915 

7 


933 


95C 
9 




N. 





1 


8 


P. ] 


P. 





625 









TABLE V 


.—LOGARITHMS OF NUMBERS. 










N. 





1 
811 


3 

828 


3 

846 


4 

863 


5 

881 


6 

898 


1 
915 


8 
933 


9 

950 


P. P. 


350 


39 794 




251 


967 


984 


*002 


*019 


*036 


*054 


*071 


*088 


*105 


*123 




252 


40 140 


157 


174 


191 


209 


226 


243 


260 


277 


295 


17 17 


253 


312 


329 346 


333 


380 


398 


415 


432 


449 


466 


.1 


1.7 


1.7 


254 


483 


500 517 


534 


551 


569 


586 


603 


620 


637 


.2 


3 


5 


3.4 


255 


654 


67i: 688 


705 


722 


739' 756 


773 


790 


807 


.3 


5 


2 


5.1 


256 


824 


841 858 


875 


892 


908 925 


942 


959 


976 


.4 


7 





6.8 


257 


993 


*010*027 


*044 


*0S1 


*077*094 


*111 


*128 


*145 


.5 


8 


7 


8.5 


258 


41 162 


179 195 


212 


229 


246 


263 


279 


296 31S 


.6 10 


5 10.2 


259 


330 


346, 383 


380 


397 


413 


430 


447 


464 


480 


.7il2 
.8 14 
.9 15 


2 11.9 
13-6 
715. 3 


360 


497 
664 


514; 530 
680 697 


J47 
714 


584 
730 


581 


597 


614 


631 


647 


261 


747 


764 


780 


797 


813 




262 


830 


846 883 


880 


896 


913 9291 946 


962 


979 




263 


995 


*012*028 


*045 


*031 


*078 *094j*lll 


*127 


*144 




264 


42 160 


177 193 


209 


226 


242 


259 


275 


292 


308 




265 


324 


341^ 357 


373 


390 


406 423 


439 


455 


472 


16 16 


266 


488 


504 521 


537 


553 


569 586 


602 


618 


635 


.1 


1-6 


1.6 


267 


651 


687 683 


700 


71C 


732 748 


765 


781 


797 


.2 


3.3 


3 


2 


268 


813 


829 846 


862 


878 


894 910 


927 


.943 


959 


.3 


4-9 


4 


8 


269 


975 


991*007 


*023 


*040 


*056 *C72 


*088 


*104 *120 


.4 
.5 
.6 
.7 


6-6 
8.2 
9-9 

11.5 


6 

8 

9 

11 


4 

6 

2 


370 


43 136 


152^ 168 


184 


200 


216 233 


249 


265 


281 






1 














271 


297 


313 329 


345 


361 


377 


3931 409 


425 


441 


.8 


13.2 


12 


8 


272 


457 


4731 489 


505 


520 


536 


552 568 


584 


600 


.9 


14.8 


14 


4 


273 


616 


632' 648 


664 


680 


695 


711 727 


743 


759 




274 


775 


791 806 


822 


833 


854 


870 886 


901 


917 




275 


933 


949 965 


980 


996 


*012 ^028 


^043 


*059 *075| 




276 


44 091 


106 122 


138 


154 


169 185 


201 


216 


232 




277 


248 


263 279 


295 


310 


326 342 


357 


373 


389 




278 


404 


420 435 


451 


467 


482 498 


513 


529 


545 


15 15 


279 


560 


576 591 


607 


622 


638 653 


669 


685 


700 


.1 
.2 
.3 
• 4 
.5 


1.5 
3.1 
4.6 


1.5 
3.0 
4.5 
6-0 
7.5 


380 


716 


731| 747 


762 


778 


793 809 


824 


839 


855 


281 


870 


886' 901 


917 


932 


948 


963 


978 


994 


*009 


282 


45 025 


040 055 


071 


086 


102 


117 


132 


148 


163 


.6 


9.3 


9.0 


283 


178 


194 209 


224 


240 


255 


270 


286 


301 


316 


.7 


10.8 


10.5 


284 


332 


347 362 


377 


393 


408 


423 


438 


454 


469 


.8 


12.4 


12.0 


285 


484 


499, 515 


530 


545 


560 


576 


591 


606 


621 


• 9 


13.9 


13.5 


28P 


636 


652 667 


682 


697 


712 


727 


743 


758 


773 




287 


788 


803! 818 


833 


848 


864 


879 


894 


909 


924 




288 


939 


9541 969 


984 


999 


*014 *029 *044 


*059 


*075 




289 


46 090 


105 120 


135 


150 


165 180 


195 


210 


225 




390 


240 


255 


269 


284 


299 


314i 329 


344 


359 


374 


.1 

.2 


2-9 


14 

1.4 
2-8 


291 


389 


404 


419 


434 


449 


464 479 


493 


508 


523 


292 


538 


553 


568 


583 


597 


612' 627 


642 


657 672 


• 3 


4.3 


4.2 


293 


687 


701 


716 


731 


746 


761 775 


790 


8051 820 


.4 


5.8 


5.6 


294 


834 


849 


864 


879 


894 


908 923 


938 


9521 967 


.5 


7.2 


7.0 


295 


982 


997 


*01I 


*026 


*041 


*055 *070 


*085 


*100*114 


.6 


8.7 


8.4 


296 


47 129 


144 


158 


173 


188 


202 


217 


232 


246 261 


.7 


10-1 


98 


297 


275 


290 


305 


319 


334 


348 


363 


378 


3921 407 


.8 


11.6 


11.2 


298 


42T 


436 


451 


465 


480 


494 


509 


523 


538: 552 


.9 


13.0 


12.6 


299 


567 
712 


581 


596 


610 


625 


639 


654 


668 


6831 697 




300 


726 

1 


741 
3 


755 
3 


770 
4 


784 
5 


799 
6 


813 

7 


828 
8 


842 
9 




N. 







P. 


I 


> 





526 



TABLE v.— LOGARITHMS OF NUMBERS. 



1 
N. 





1 

726 
871 


2 


3 


4 

770 


5 

784 


6 


7 


8 
828 


9 

842 


P. P. 


300 


47 712 
856 


741 


755 


799 


813 




301 


885 


900 


914 


928 


943 


957 


972 


986 




302 


48 000 


015 


029 


044 


058 


072 


087 


101 


115 


130 




303 


144 


158 


173 


187 


201 


216 


230 


244 


259 


273 




304 


287 


301 


316 


330 


344 


358 


373 


387 


401 


415 




305 


430 


444 


458 


472 


487 


501 


515 


529 


543 


558 




306 

,307 

308 


572 
714 
855 


586 
728 
869 


600 
742 
883 


614 
756 
897 


629 
770 
911 


643 
784 
925 


657 
798 
939 


671 
812 
953 


685 
827 
967 


699 
841 
982 


•1 

• 2 
.3 
.4 
.5 
.6 

• 7 
8 

.9 


1? 

1.4 

r> f\ 


14 

1.4 


309 


996 


*010 


*024 


*038 


*052 


=*=066 


*080 


*094 


*108 


*122 
262 


2 

4 

5 

7 

8 

10 

11 

13 


8 

• 2 
7 

• I 


2 
4 
5 
7 
8 
9 
11 
12 




2 


jaio 

! 


49 136 


150 


164 


178 


192 


206 


220 


234 


248 


•6 



311 


276 


290 


304 


318 


332 


346 


359 


373 


387 


401 


4 

• 8 

• 2 
8 


312 
313 
314 


415 
554 
693 


429 
568 
707 


443 
582 
729 


457 
596 
734 


471 
610 
748 


485 
624 
762 


499 
637 
776 


513 
651 
789 


526 
665 
803 


540 
679 
817 


315 


831 


845 


858 


872 


886 


900 


913 


927 


941 


955 




316 


968 


982 


996 


*010 


*023 


*037 


^051 


*065 


-078 


=^092 




317 


50 106 


119 


133 


147 


160 


174 


188 


201 


215 


229 




318 


242 


256 


270 


283 


297 


311 


324 


338 


352 


365 




319 


379 


392 


406 


420 


433 


447 


460 


474 


488 


501 
637 
77? 




320 


515 


528 


542 


555 


569 


583 


596 
731 


610 
745 


623 
758 




321 


650 


664 


677 


691 


704 


718 


.1 
.2 
■ 3 
■4 

• 5 

• 6 
•7 
.8 
.9 


13. 13 


322 


785 


799 


812 


826 


839 


853 


866 


880 


893 


007 


1 

2 

4 

5. 

6. 

8. 

9. 
10. 
12. 


6 1 

01 3 


6 


323 


920 


933 


947 


960 


974 


987 


*001 


*014 


*027 


*041 


6 
9 
2 
5 
8 
1 


324 


51 054 


068 


081 


094 


108 


121 


135 


148 


161 


175 


325 


188 


201 


215 


228 


242 


255 


288 


282 


295 


308 


326 
327 
328 


322 
455 
587 


335 
468 
600 


348 
481 
614 


361 
494 
627 


375 
508 
640 


388 

521 
653 


401 
534 
667 


415 
547 
680 


428 
561 
693 


441 
574 
700 


7 

1 

4 
8 

T 




7 
9 

1 r\ 


329 


719 


733 


746 


759 


772 


785 


798 


812 


825 


838 


10-4 
11.7 


330 


85l 


864 


877 


891 


904 


917 


930 


943 


956 


969 
*100 




331 


983 


996 


*009 


*022 


*035 


*048 


*06l 


*074 


*087 




332 


52 114 


127 


140 


153 


166 


179 


102 


205 


218 


231 




333 


244 


257 


270 


283 


296 


309 


322 


335 


348 


361 




334 


374 


387 


400 


413 


426 


439 


452 


465 


478 


491 




335 


504 


517 


530 


543 


556 


569 


582 


595 


608 


62] 




336 


634 


647 


660 


672 


685 


698 


711 


724 


737 


750 





337 
338 


763 
89l 


776 
904 


789 
917 


801 
930 


814 
943 


827 
956 


840 
968 


853 
981 


868 
994 


879 
*007 


•1 
.2 
.3 

• 4 

• 5 

• 6 
-7 
.8 
.9 


L'4 

1.2 


1'^ 

1.2 


339 


53 020 


033 


045 


058 
186 


071 
199 


084 


097 


109 


122 


135 


2 
3 
5 
6 
7 
8 
10 


y 
7 

2 
5 
7 
Q 


2 
3 
4 
6 
7 
8 
9 


4 
6 


340 


148 

275 
402 


160 

288 

415 


173 


211 


224 


237 


250 


262 

390 
516 


8 



341 

342 


301 
428 


313 
440 


326 
453 


339 
466 


352 
478 


364 
491 


377 
504 


2 
4 


343 


529 


542 


554 


567 


580 


592 


605 


618 


630 


6^3 


6 


344 


656 


668 


681 


693 


706 


719 


731 


744 


756 


769 


11 z 


lO-o 


345 


782 


794 


807 


819 


832 


845 


857 


870 


882 


895 




346 


907 


920 


932 


945 


958 


970 


983 


995 


*008 


*020 




347 


54 033 


045 


058 


070 


083 


095 


108 


120 


133 


145 




348 


158 


170 


183 


195 


208 


220 


232 


245 


257 


270 




349 


282 


295 


307 
431 

3 


320 
444 

3 


332 
456 

4 


344 


357 


369 


382 


394 
5li 

9 




350 


407 



419 
1 


469 
5 


481 
6 


493 

7 


506 
8 




N. 


P. P. 



52; 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 

419 


2 

431 


3 

444 


4 

456 


5 

469 


6 

481 


7 
493 


8 
506 


9 

518 


P 


. p. 




350 


54 407 




12 




























351 


530 


543 


555 


561 


580 


592 


605 


617 


629 


642 


.1 


1.2 




352 


654 


666 


679 


691 


703 


716 


728 


740 


753 


765 


.2 


2 


5 




353 


777 


790 


802 


814 


826 


839 


851 


863 


876 


888 


-3 


3 


7 




354 


900 


912 


925 


937 


949 


961 


974 


986 


998 


*010 


.4 


5 







355 


55 023 


035 


047 


059 


071 


084 


096 


108 


120! 133 


.5 


6 


2 




35S 


145 


157 


169 


181 


194 


206 


218 


230 


242 


254 


.6 


7 


5 




357 


267 


279 


291 


303 


315 


327 


340 


352 


364 


376 


• 7 


8 


7 




358 


388 


400 


412 


424 


437 


449 


461 


473 


485 


497 


.8 


10 






359 


509 
630 

750 


521 
642 
762 


533 
654 


545 
666 
787 


558 
678 
799 


570 


582 


594 


606 


618 


.9 
.1 


11 


2 




360 


690 


702 


714 


726 


738 


13 

1.2 




361 


775 


811 


823 


835 


847 


859 




382 


871 


883 


895 


907 


919 


931 


943 


955 


966 


978 


.2 


2 


4 




363 


990 


*002 


*014 


*026 


*038 


*050 


*062 


*074 


*086 


*098 


.3 


3 


6 




364 


56 110 


122 


134 


146 


158 


170 


181 


193 205 


217 


• 4 


4 


8 




365 


229 


241 


253 


265 


277 


288 


300 


312 


324 


336 


.5 


6 







366 


348 


360 


372 


383 


395 


407 


419 


431 


443 


455 


.6 


7 


2 




387 


466 


478 


490 


502 


514 


525 


537 


549 


561 


573 


.7 


8 


4 




368 


585 


596 


608 


620 


632 


643 


655 


667 


679 


691 


.8 


9 


6 




s-sa 


702 
820 


714 
832 


726 
843 


738 
855 


749 
867 


761 
879 
996 


773 
890 

*007 


785 


796 


808 


.9 
.1 


10. 8 




370 


902 


914 


925 
*042 




371 


937 


949 


961 


972 


984 


*019 


*031 




372 


57 054 


066 


077 


089 


101 


112 


124 


136 


147 


159 


.2 


2 


3 




373 


171 


182 


194 


206 


217 


229 


240 


252 


264 


275 


.3 


3 


4 




374 


287 


299 


310 


322 


333 


345 


357 


368 


380 


391 


• 4 


4 


6 




375 


403 


414 


426 


438 


44Q 


461 


472 


484 


495 


507 


.5 


5 


7 




376 


519 


530 


542 


553 


565 


576 


588 


599 


611 


622 


.6 


6 


9 




377 


634 


645 


657 


668 


680 


691 


703 


714 


726 


737 


.7 


8 







378 


749 


760 


772 


783 


795 


806 


818 


829 


841 


852 


.8 


9 


2 




379 


864 


875 


887 


898 


909 

*024 

138 


921 


932 


944 


955 


967 


.9 
.1 


10.3 

11 

1.1 




380 


978 


990 


*001 


*012 


*035 


*047 


*058 


*069 


*C81 




381 


58 092 


104 


115 


126 


149 


161 


172 


183 


195 




382 


206 


217 


229 


240 


252 


263 


274 


286 


297 


308 


.2 


2 


2 




383 


320 


331 


342 


354 


365 


376 


388 


399 


410 


422 


• 3 


3 


3 




384 


433 


444 


455 


467 


478 


489 


501 


512 


523 


535 


.4 


4 


4 




385 


546 


557 


568 


580 


591 


602 


613 


625 


636 


647 


.5 


5 


5 




386 


658 


670 


681 


692 


703 


715 


726 


737 


748 


760 


.6 


6 


6 




387 


771 


782 


793 


804 


816 


827 


838 


849 


861 


872 


.7 


y 


7 




388 


883 


894 


905 


916 


928 


938 


950 


961 


972 


984 


• 8 


8 


8 




389 


995 


*006 


*017 


*028 


*039 


*050 


*ce2 


*073 


-■*'G84 


*095 
206 


-9 

.1 


9.9 

^% 

1.0 




390 


59 106 


117 


128 


140 


151 

262 


162 


173 


184 


195 




391 


217 


229 


240 


251 


273 


284 


295 


306 


317 




392 


328 


339 


351 


362 


373 


384 


395 


4C6 


417 


428 


.2 


2 


1 




393 


439 


450 


46] 


472 


483 


494 


505 


516 


527 


538 


• 3 


3 


1 




394 


549 


560 


571 


582 


593 


604 


615 


626 


637 


648 


.4 


4 


2 




395 


659 


670 


681 


692 


703 


714 


725 


736 


747 


758 


.5 


5 


2 




39R 


769 


780 


791 


802 


813 


824 


835 


846 


857 


868 


.6 


6 


3 




397 


879 


890 


901 


912 


923 


933 


944 


955 


966 


97^ 


• 7 


7 


3 




398 


988 


999 


*010 


*021 


*032 


*043 


*053 


*064 


*075 


*C86 


.8 


8 


4 




399 


60 097 


108 


119 


130 


141 


151 


162 


173 


184 


195 


.9 


9 4 




400 


206 


217 


227 


238 
3 


249 
4 


260 
5 


271 
6 


282 

7 


293 
8 


303 
9 




N. 





1 


2 


r 


.P 







528 









TABLE v.— LOGARITHMS OF NUMBERS. 






N. 





1 


2 


3 

238 


4 


5 


6 

271 


7 
282 


8 

293 
401 


9 

303 
412 


P.P. 


400 


60 206 


217 


1 
227| 


249 


280 




401 


314 


325 


336l 


347 


357 


368| 


3791 390 




402 


422 


433 


444 


455 


466 


476i 487i 498 


509 


519 




403 


530 


541 


552 


563 


573 


584! 595 606 


616 


627 


11 


404 


638 


649 


659: 


670 


681 


6921 702 713 


724 


735 


.1 


1.1 


405 


745 


756 


7671 


111 


788 


799 810 820 


831 


842 


• 2 


2- 


2 


408 


852 


863 874 


884 


895 


9061 916' 927 


^938 


949 


.3 


3 


3 


407 


959 


970 981' 


991i*002l 


*013|*023 *034| 


*044 


*055 


•4 


4 


4 


408 


61 066 


076 


087i 098; 108! 


119 130! 


140 


151 


161 


.5 


5 


5 


409 


172 


183 


193 


204| 


215 


225 


236, 


246 


257 


268 


.6 
.7 
.8 

Q 


6 
7 
8 


6 
7 
8 


410 


278 


289 


299 


310 


320 


33l 


342| 


352 


363 
468 


373 
479 


411 


384 


394 


405 


416 


426 


437 


447 j 


458 


• a . vj 


412 


489 


500 


511 


5211 5321 


542 


553 


563 


574 


584 




413 


595 


605 


616 626i 637 


647 


658i 


668 


679 


689 




^14 


700 


710 


721i 731! 742 


752 


763; 773 


784 


794 




415 


805 


815 


825! 8361 846 


857 


867i 878 


888 


899 


10 


416 


909 


920 


930l 940 


951 


961 


972 982 


993 


*003 


• 1 


1.0 


417 


62 013 


024 


0341 045 


055 


065 


076' 086 


097 


107 


.2 


2 


1 


418 


117 


128 


138| 149 


159 


139 


180 190 


200 


211 


.3 


3 


1 


419 


22l 


232 


242 252 263 


273 


283| 294 


304 


314 


.4 
.5 
.6 
.7 
.8 


4 
5 
6 
7 
8 


2 
2 

4 


430 


325 


335 


345 


356 


366 


376 


387 


397 


407 


418 


421 


428 


438 


449! 459 489 


480 


490: 500 


510 


521 


422 


531 


541 


552 


582i 572 


5821 593 603 


613 


624 


.9 


9 


4 


423 


634 


644 


654 


665! 675 


685 695' 706 


716 


726 




424 


736 


747 


757 


767| 777 


788 


798! 808 


818 


828 




425 


839 


849 


859 


869 879 


890 


900 910 


920 


931 




428 


941 


951 


96l| 9711 981 


992 


*002;*012 


*022 


*032 




427 


63 043 


053 


063 


073| 083 


093 


104 


114 


124 


134 


10 


428 


144 


154 


164 


1751 185 


195' 


205 


215 


225 


235 


• 1 


1.0 


429 


245 


256 


266 


276 286 


296 


306 


316 


326 


336 


.2 
.3 
.4 
.5 
.6 


2 
3 
4 
5 
6 


-0 
.0 
.0 
-0 
.0 


430 


347 


357 


367 


377 387 


397 


407 


417 


427 


437 
538 


431 


447 


458 


468 


478i 488 


498 


508 


518 


528 


432 


548 


558 


568 


578 


588 


598 


608 


618 


628 


639 


.7 


7 





433 


649 


659 


669 


679 


689 


699 


709 


719 


729 


739 


.8 


8 


.0 


434 


749 


759 


769 


779 


789 


799 


809 


819 


829 


839 


.9 


9 





435 


849 


859 


869 


879 


889 


899 


909 


9191 928 


938 




438 


948 


958 


968 


978 


988 


998 


*008 


*018l*028 


*038 




437 


64 048 


058 


068 


078 


088 


098 


107 


117 


127 


137 




438 


147 


157 


167 


177 


187 


197 


207 


217 


226 


236 




439 


246 
345 


256 


266 


276| 286 


296 


306 
404 


315 
414 


325 
424 


335 
434 


.1 
.2 
.3 
.4 


0^9 
19 
28 
3.8 


440 


355 


365 375 


384 


394 


441 


444 


453 


463 473 


483 


493 


503 


512 


522 


532 


442 


542 


552 


562 57l 


581 


591 


601 


611 


621 


630 


.5 


4.7 


443 


640 


650 


660 


1 670 


679 


689 


699 


' 709 


718i 728 


.6 


5.7 


444 


738 


748 


758 


1 767 


! 777 


787 


797 


1 806 


8161 826 


.7 


6.6 


445 


836 


846 


855 


8651 875 


885 


894 


904 


914; 923 


.8 


7.6 


448 


933 


943 


953 


962 


972 


982 


992 


*001 


*011 *021 


.9 


8.5 


447 


65 031 


040 


050; 060 


069 


079 


089 


098 


108 118 




448 


128 


137 


147 


157 


166 


176 


186 


195 


i 205! 215 




449 


224 
32l 


234 


244 


: 253 


263 


273 


j 282 


292 


1 302| 311 




450 


331 


34C 
2 


35C 
3 


36C 
4 


369 

5 


^ 379 
6 


389 

7 


398 
8 


408 
9 




N. 





1 


P.P. 



529 









TABLE V 


.—LOGARITHMS ( 


3F NUMBERS. 




N. 





1 


2 


3 

350 

446 


4 

360 


5 


6 


7 


8 
398 


9 

408 


P. P. 


450 


65 32l 
417 


331 
42^ 


340 
437 


369 


379 


389 




451 


456 


466 


475 


485 


494 


504 




452 


514 


523 


533 


542 


552 


562 


571 


581 


590 


600 


10 


453 


610 


619 


629 


638 


648 


657 


667 


677 


686 


696 


.1 


1-0 


454 


705 


715 


724 


734 


744 


753 


763 


772 


782 


791 


.2 


2.0 


455 


801 


810 


820 


830 


839 


849 


858 


868 


877 


887 


-3 


30 


456 


896 


906 


915 


925 


934 


944 


9C3 


963 


972 


982 


• 4 


4.0 


457 


991 


*001 


*010 


*02C 


*029 


=^■039 


*048 


*C58 


*C67 


*077 


5 


5.0 


458 


66 086 


096 


105 


115 


124 


134 


143 


153 


162 


172 


6 


6.0 


459 


181 


190 


200 


208 


219 


228 


238 


247 


257 


266 


.7 

.8 

9 


7.0 


460 


276 
370 


285 


294 


304 


313 


323 


332 


342 


351 


360 


80 
9.0 


461 


379 


389 


398 


408 


417 


426 


436 


445 


455 




462 


464 


473 


483 


492 


502 


511 


520 


530 


539 


548 




463 


558 


567 


577 


588 


595 


605 


614 


623 


633 


642 




464 


652 


661 


670 


680 


689 


698 


708 


717 


726 


736 




465 


745 


754 


764 


773 


782 


792 


801 


810 


820 


829 


9 


466 


838 


848 


857 


86G 


876 


885 


894 


904 


913 


922 


•1 


0.9 


467 


931 


941 


950 


959 


969 


978 


987 


996 


*C06 


^015 


.2 


1.9 


468 


67 024 


034 


043 


052 


061 


071 


080 


089 


099 


108 


• 3 


2.8 


469 


117 


126 


136 


145 


154 


163 


173 


182 


191 


200 


.4 
.5 


!1 
















4-7 


470 


210 
302 


219 


228 


237 

329 


246 


256 


265 


274 


283 


293 


.6 

• 7 

8 


76 


471 


311 


320 


339 


348 


357 


366 


376 


385 


472 


394 


403 


412 


422 


431 


44C 


449 


458 


467 


477 


• 9 


8.5 


473 


486 


495 


504 


513 


523 


532 


541 


550 


559 


568 




474 


578 


587 


596 


6C5 


614 


623 


633 


642 


651 


660 




475 


689 


678 


687 


697 


706 


715 


724 


733 


742 


751 




476 


760 


770 


779 


788 


797 


806 


815 


824 


833 


842 




477 


852 


861 


870 


879 


888 


897 


906 


915 


924 


933 




478 


943 


952 


961 


970 


97G 


988 


997 


*CC6 


*G15 


*024 


9 


479 


68 033 


042 


051 


060 


070 


079 


088 


C97 


106 


115 


.1 
.2 

• 3 

• 4 

• 5 


0.9> 
18 
2.7 
3-6 
4-5 


480 


124 


133 


142 


151 


160 


169 


178 


187 


196 


205 


481 


214 


223 


232 


241 


250 


259 


268 


277 


286 


295 


482 


304 


313 


322 


331 


340 


349 


358 


367 


376 


385 


.6 


5.4 


483 


394 


403 


412 


421 


430 


439 


448 


457 


466 


475 


• 7 


6.3 


484 


484 


493 


502 


511 


520 


529 


538 


547 


556 


565 


.8 


7.2 


485 


574 


583 


592 


601 


610 


619 


628 


637 


646 


654 


• 9 


8.1 


486 


663 


672 


681 


690 


699 


708 


717 


726 


735 


744 




487 


753 


762 


770 


779 


788 


797 


806 


815 


824 


833 




488 


842 


851 


860 


868 


877 


886 


895 


904 


913 


922 


*^w« 


489 


931 
69 019 


940 
028 


948 


957 


966 


975 


984 


993 


*002 


'^010 


M 


490 


037 


046 


055 


064 


073 


081 


090 


099 


.1 

• 2 


o^g 1 

1.7 1 


491 


108 


117 


126 


134 


143 


152 


16] 


170 


179 


187 


492 


196 


205 


214 


223 


232 


240 


249 


258 


267 


276 


.3 


2.5 


493 


284 


293 


302 


311 


320 


328 


337 


346 


355 


364 


■4 


34 


494 


372 


381 


390 


399 


4C8 


416 


425 


434 


443 


451 


.5 


4.2 


495 


460 


469 


478 


487 


495 


504 


51? 


522 


530 


539 


.6 


5.1 


496 


548 


557 


565 


574 


583 


502 


6CC 


609 


618 


627 


.7 


5.9 


497 


635 


644 


653 


662 


670 


679 


688 


697 


705 


71^ 


.8 


6-8 


498 


723 


731 


7-40 


749 


.758 


76e 


775 


784 


792 


8C] 


9 


7.6 1 


499 


810 
897 


819 
905 

1 


827 


836 


845 


853 


862 


871 


879 


888 
975 

9 


] 


600 


914 
2 


923 
3 


93l 
4 


940 
5 


949 
6 


958 

7 


966 
8 


J 


N. 





P. 


P. 



530 







TABLE V 


.— LC 


)GAI 


IITH 


MS OF NUMBERS. 




N. 





1 
905 


2 

914 


3 

923 


4 

931 


5 

940 


6 

949 


7 

958 


8 
966 


9 

975 


P. P. 


500 


69 897 




501 


984 


992 


*001 


*010 


*018 


"^027 


*036 


*044 


*053 


*06T 




502 


70 070 


079 


087 


096 


105 


113 


122 


131 


139 


148 


9 


503 


157 


165 


174 


182 


191 


2CC 


208 


217 


226 


234 


.1 


0.9 


504 


243 


251 


260 


269 


277 


286 


294 


303 


312 


320 


• 2 


18 


505 


329 


337 


346 


355 


363 


372 


380 


389 


398 


406 


.3 


2.7 


506 


415 


423 


432 


44] 


449 


458 


466 


475 


483 


492 


.a 


36 


507 


501 


509 


'518 


526 


535 


543 


552 


560 


569 


578 


• 5 


4-5 


508 


586 


595 


603 


612 


620 


629 


637 


646 


654 


66£ 


.6 


5-4 


509 


672 
757 
842 


680 


689 


697 


706 


714 


723 


731 


740 


7U 


•7 
.8 
.9 


6.3 
7.2 
8.1 


510 


765 
850 


774 


782 


791 


799 


808 


816 


825 


833 
918 


511 


859 


867 


876 


884 


893 


901 


910 




512 


927 


935 


944 


952 


961 


969 


978 


986 


995 


^003 




513 


71 Oil 


020 


028 


037 


045 


054 


062 


071 


079 


088 




514 


096 


105 


113 


121 


130 


138 


147 


155 


164 


172 




515 


180 


189 


197 


206 


214 


223 


231 


239 


248 


256 


8 


516 


265 


273 


282 


290 


298 


307 


315 


324 


332 


340 


.1 


0.8 


517 


349 


35*7 


366 


374 


382 


391 


399 


408 


416 


424 


.2 


1-Z 


518 


433 


441 


449 


458 


466 


475 


483 


491 


500 


508 


.3 


2.5 


519 


516 


525 


533 


542 


550 


558 


567 


575 


583 


592 


•4 
.5 
.6 
.7 
.8 


3.4 
4.2 
5.1 
5.9 
68 


530 


600 


608 


617 


625 


633 


642 


650 


659 


667 


675 


521 


684 


692 


700 


709 


717 


725 


734 


742 


750 


758 


522 


767 


775 


783 


792 


800 


808 


817 


825 


833 


84? 


.9 


7.6 


523 


850 


858 


867 


875 


883 


891 


900 


908 


916 


925 




524 


933 


941 


949 


958 


966 


974 


983 


991 


999 


*007 




525 


72 016 


024 


032 


040 


049 


057 


065 


074 


082 


090 




526 


098 


107 


115 


123 


131 


140 


148 


156 


164 


173 




527 


181 


189 


197 


206 


214 


222 


230 


238 


247 


255 




528 


263 


271 


280 


288 


296 


304 


312 


321 


329 


337 


8 


529 


345 


354 


362 


370 


378 


386 


395 


403 


411 


419 


• 1 

.2 
.3 

• 4 


0.8 

1.6 
2.4 
3.2 


530 


427 


436 


444 


452 


46G 


468 


476 


485 


493 


501 
























531 


509 


517 


526 


534 


542 


550 


558 


566 


575 


583 


.5 


4.0 


532 


591 


599 


607 


615 


624 


632 


640 


648 


656 


664 


.6 


48 


533 


672 


681 


689 


697 


705 


713 


721 


729 


738 


74P 


•7 


5-6 


534 


754 


762 


770 


778 


786 


795 


803 


811 


819 


827 


.8 


6.4 


535 


835 


843 


851 


859 


868 


876 


884 


892 


90C 


908 


.9 


7.2 


536 


916 


924 


932 


941 


949 


957 


965 


973 


981 


989 




537 


997 


*005 


*013 


*021 


*030 


*038 


*046 


*054 


*062 


*070 




538 


73 078 


086 


094 


102 


110 


118 


126 


134 


143 


151 




539 


159 
239 


167 


175 


183 


191 


199 


207 


215 


223 


231 




540 


247 


255 


263 


271 


279 


287 


^95 


303 


311 


.1 


o'. 
























541 


319 


328 


336 


344 


352 


360 


368 


376 


384 


392 


.2 


1.5 


542 


400 


408 


416 


424 


432 


440 


448 


456 


464 


472 


.3 


2.2 


543 


480 


488 


496 


504 


512 


520 


528 


536 


544 


552 


• 4 


3.0 


544 


560 


568 


576 


58^ 


592 


600 


608 


615 


623 


631 


.5 


3-7 


545 


639 


647 


655 


663 


671 


679 


687 


695 


703 


711 


.6 


4.5 


546 


719 


727 


735 


743 


751 


759 


767 


775 


783 


791 


.7 


5.2 


547 


798 


806 


814 


822 


830 


838 


846 


854 


862 


870 


.8 


6 


548 


878 


886 


894 


902 


909 


917 


925 


933 


941 


949 


9 


6.7 


549 


957 


965 


973 


981 


989 


997 


*004 


*012 


*020 


*028 




550 


74 036 


044 


052 


060 


068 


075 
5 


083 
6 


091 

7 


099 
8 


107 




N. 





1 


2 


3 


4 


9 


P. P, 



531 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 

083 


7 


8 


9 

107 


P. P. 


550 


74 036 


044 


052 


060 


068 


075 


09l 


099 




551 


115 


123 


131 


139 


146 


154 


162 


170 


178 


186 




552 


194 


202 


209 


217 


225 


233 


241 


249 


257 


264 




553 


272 


230 


288 


296 


3G4 


312 


319 


327 


335 


343 




554 


351 


359 


386 


374 


382 


390 


398 


406 


413 


421 




555 


429 


437 


445 


453 


460 


468 


476 


484 


492 


499 




556 


507 


515 


523 


531 


538 


546 


554 


562 


570 


577 




557 


585 


593 


601 


609 


616 


624 


632 


640 


648 


655 


.1 
.2 
-3 
.4 

• 5 

• 6 
•7 
.8 
.9 


» 


558 


663 


671 


679 


687 


094 


702 


710 


718 


725 


733 


" 

1 

2 

3 

4 

4 

5 

6 

7 


•0 

6 

4 


559 


741 


749 


756 


764 


772 


780 
857 


788 


795 


803 


811 


560 


819 


826 


834 


842 


850 


865 


873 


881 


888 


• 2 



561 


896 


904 


912 


919 


927 


935 


942 


950 


958 


966 


8 

6 

•4 

• 2 


562 
563 
564 


973 

75 051 

128 


981 
058 
135 


989 
086 
143 


997 
074 
151 


*004 
081 
158 


*012 
089 
166 


*020 
097 
174 


*027 
105 
182 


*035 

112 
189 


*043 
120 
197 


565 


205 


212 


220 


228 


235 


243 


251 


258 


266 


274 




566 


281 


289 


297 


304 


312 


320 


327 


335 


343 


350 




567 


358 


368 


373 


381 


389 


396 


404 


412 


419 


427 




568 


435 


442 


450 


458 


485 


473 


480 


488 


496 


50S 




569 


511 


519 


526 


534 


541 


549 


557 


564 


572 
648 


580 

656 




570 


587 


595 


602 


610 


618 


625 


633 


641 




571 
572 
573 


683 
739 
815 


671 
747 
823 


679 
755 
830 


686 
762 
838 


694 
770 
846 


70l 
777 
853 


709 
785 
861 


717 
792 
868 


724 
800 
876 


732 
808 
883 


.1 

.2 
.3 
.4 

• 5 

• 6 

• 7 
8 

.9 




574 


891 


899 


906 


914 


921 


929 


936 


944 


951 


959 


2 
3 
3 
4 
5 
6 
6 



7 
5 
2 

7 


575 


967 


974 


982 


989 


997 


=^004 


*012 


*019 


='=027 


*034 


576 


76 042 


050 


057 


065 


072 


080 


087 


095 


102 


110 


577 
578 
579 


117 
193 
268 


125 
200 
275 


132 
208 
283 


140 
215 
290 


147 
223 
298 


155 
230 
305 


162 
238 
313 


170 
245 
320 


178 
253 
328 


185 
260 
335 


580 


343 


350 


358 


365 


372 
447 


380 


387 


395 


402 


410 




581 


417 


425 


432 


440 


455 


462 


470 


477 


485 




582 


492 


500 


507 


514 


522 


529 


537 


544 


552 


559 




583 


567 


574 


582 


589 


596 


604 


611 


619 


626 


634 




584 


641 


648 


656 


663 


671 


678 


686 


693 


700 


708 




585 


715 


723 


730 


738 


745 


752 


760 


767 


775 


782 




586 


790 


797 


804 


812 


819 


827 


834 


841 


849 


866 




587 


864 


871 


878 


888 


893 


901 


908 


915 


923 


930 


.] 

.2 

• 3 
.4 
.5 

• 6 
.7 
.8 
.9 




588 


937 


945 


952 


960 


967 


974 


982 


989 


997 


*0C4 


1. 
2. 
2. 
3. 
4. 
4. 
5. 
6. 


/ 
4 
1 


589 


77 Oil 


019 


026 


033 
107 
181 


041 
114 
188 


048 


055 
129 


063 
136 


070 


078 


590 


085 
158 


092 


100 


122 


144 


151 


8 
5 


591 


166 


173 


195 


203 


210 


217 


225 


2 
9 
6 
3 


592 
593 
594 


232 
305 
378 


239 
313 
386 


247 
320 
393 


254 
327 
400 


261 
335 
408 


269 
342 
415 


276 
349 
422 


283 
356 
430 


291 
364 
437 


29P 
371 
444 


595 


451 


459 


466 


473 


481 


488 


495 


503 


510 


517 




596 


524 


532 


539 


546 


554 


561 


568 


575 


583 


590 




597 


597 


604 


612 


619 


626 


634 


641 


648 


655 


663 




598 


670 


677 


684 


692 


699 


706 


713 


721 


728 


735 




599 


742 


750 


757 


764 


771 


779 


786 


793 


800 


808 




600 


815 


822 

1 


820 


837 
3 


844 
4 


851 


858 


866 

7 


873 

8 


880 
9 




N. 





2 


5 


6 


P. P. 



532 







TABLE V 


—LOGARITHMS OF NUMBERS. 






N. 





1 


2 

829 


3 

837 


4 

844 

916 


5 

851 


6 

858 


7 
866 


8 

873 

945 


9 

880 
952 


P 


. P. 


600 


77 815 
887 


822 
894 






601 


902 


909 


923 


931 


938 




602 


959 


967 


974 


981 


988 


995 


*003 


*010 


*017 


*024 






603 


78 031 


039 


046 


053 


060 


067 


075 


082 


089 


096 






604 


103 


111 


118 


125 


132 


139 


147 


154 


161 


168 






605 


175 


182 


190 


197 


204 


211 


218 


226 


233 


240 






606 


247 


254 


281 


269 


276 


283 


290 


297 


304 


311 




_ 


607 
608 
609 


319 
390 
461 

533 

604 


326 
397 
469 

540 

611 


333 

404 
476 

547 

618 


340 
412 
483 


347 
419 
490 


354 
426 
497 


362 
433 
504 


369 
440 
511 


376 
447 
518 


383 

454 
526 


.1 
.2 
.3 
.4 
.5 
.6 
• 7 
.8 
.9 




1 
2 
3 
3 
4 
5 

e 

6 


■ 5 
.2 


610 


554 
625 


561 
632 


568 


575 


583 


590 
661 


597 
668 


.0 
.7 


611 


639 


646 


654 




612 


675 


682 


689 


696 


703 


710 


717 


725 


732 


739 


•Q 

•7 


613 

614 


746 
817 


753 
824 


760 
831 


767 
838 


774 
845 


781 
852 


788 
859 


795 
866 


802 
873 


810 
880 


615 


887 


894 


901 


908 


915 


923 


930 


937 


944 


951 






616 


958 


965 


972 


979 


986 


993 


*000 


*007 


*014 


*021 






617 


79 028 


035 


042 


049 


056 


063 


070 


078 


085 


092 






618 


099 


108 


113 


120 


127 


134 


141 


148 


155 


162 






619 


169 


176 


183 


190 


197 


204 


211 


218 


225 


232 






630 


239 


246 


253 


260 


267 


274 
344 


281 

351 


288 
358 


295 
365 


302 
372 


.1 
• 2 
.3 
.4 
.5 
.6 
•7 
.8 
.9 




621 


309 


316 


323 


330 


337 


7 


622 


379 


386 


393 


400 


407 


414 


421 


428 


435 


442 


u 

1 
2 
2 
3 
4 
4 
5 
6 


/ 
4 
1 
8 
5 
2 
9 
6 
3 


623 
624 
625 


449 
518 
588 


456 
525 
595 


462 
532 
602 


469 
539 
609 


476 
546 
616 


483 
553 
622 


490 
560 
629 


497 
567 
636 


504 
574 
643 


511 
581 
650 


626 
627 
628 
629 


657 
727 
796 
865 


664 
733 
803 
872 


671 
740 
810 
879 


678 
747 
816 
886 


685 
754 
823 
892 


692 
761 
830 
899 


699 
768 
837 
906 


706 
775 
844 
913 


713 
782 
851 
920 


720 
789 
858 
927 


630 


934 


941 


948 


954 


96l 


968 


975 


982 


989 


996 




631 


80 003 


010 


016 


023 


030 


037 


044 


051 


058 


065 




632 


071 


078 


085 


092 


099 


106 


113 


120 


126 


133 






633 


140 


147 


154 


161 


168 


174 


181 


188 


195 


202 






634 


209 


216 


222 


229 


236 


243 


250 


257 


263 


270 






635 


277 


284 


291 


298 


304 


311 


318 


325 


332 


339 


• 




636 


345 


352 


359 


366 


373 


380 


386 


393 


400 


407 




0^- 


637 


414 


421 


427 


434 


441 


448 


455 


461 


468 


475 


.1 
.2 

• 3 
.4 
.5 

• 6 
.7 
.8 

• 9 


638 


482 


489 


495 


502 


509 


516 


523 


529 


536 


543 




1 

1 

2. 

3 

3. 

4 

5 


o 
3 
9 


639 


550 


557 


563 


570 


577 


584 


591 


597 


604 


611 


640 


618 


625 


631 


638 


645 


652 


658 


6d5 


672 


679 




641 


686 


692 


699 


706 


713 


719 


726 


733 


740 


746 


1 


642 


753 


760 


767 


774 


780 


787 


794 


801 


807 


814 


643 


821 


828 


834 


841 


848 


855 


861 


868 


875 


882 


644 


888 


895 


902 


909 


915 


922 


929 


936 


942 


949 


5 o 


645 


956 


962 


969 


976 


983 


989 


996 


*003 


*010 


*016 






646 


81 023 


030 


036 


043 


050 


057 


063 


070 


077 


083 






647 


090 


097 


104 


110 


117 


124 


130 


137 


144 


151 






648 


157 


164 


171 


177 


184 


191 


197 


204 


211 


218 






649 


224 


231 


238 


244 


251 


258 


264 


271 


278 


284 






650 


29l 


298 

1 


304 


311 


318 
4 


324 
5 


331 
6 


338 

7 


345 
8 


35l 
9 




N. 





2 


3 


P 


P. 



533 







r 


FABLE V 


.—LOGARITHMS OF NUMBERS. 






N. 





1 


2 

304 


3 

311 


4 

318 
385 


5 

324 
391 


6 

33l 
398 


7 

338 

405 


8 

345 
411 


9 



351 
418 


P 


. P. 


650 


81 29l 


298 






651 


358 


365 


371 


378 




652 


425 


431 


438 


444 


451 


458 


464 


471 


478 


484 






653 


491 


498 


504 


511 


518 


524 


531 


538 


54^. 


551 






654 


558 


564 


571 


577 


584 


591 


597 


604 


611 


617 






655 


624 


631 


637 


644 


650 


657 


664 


670 


677 


684 






656 


690 


697 


703 


710 


717 


723 


730 


736 


743 


75C 






657 
658 
659 


756 
822 
888 


763 
829 
895 


770 
836 
901 


776 
842 
908 


783 
849 
915 

980 


789 
855 
92i 

987 


796 
862 
928 

994 


805 
869 

934 

*OGC 


809 
875 
941 


816 
882 
948 


• 1 
.2 
.3 
.4 
-5 
.6 

• 7 
8 

• 9 



1 
2 
2 
3 
4 
4 
5 
6 


7 

4 

.1 


660 


954 


961 


967 

033 
099 
164 
230 


974 


*007 


*013 


8 

• 5 


661 
662 
663 
664 


82 020 
086 
151 
217 


026 
092 
158 
223 


040 
105 

171 
236 


046 
112 
177 
^43 


053 
118 
184 

249 


059 
125 
190 
256 


066 
131 

197 
262 


072 
138 
203 
269 


079 
145 
210 
275 


• 2 
9 
6 

• 3 


665 


282 


288 


295 


302 


308 


315 


321 


328 


334 


341 






666 


347 


354 


360 


367 


373 


38C 


386 


393 


399 


406 






667 


412 


419 


425 


432 


438 


445 


451 


458 


464 


471 






668 


477 


484 


490 


497 


503 


510 


516 


523 


529 


536 






669 


542 
607 


549 


555 


562 


568 


575 


581 


588 


594 


601 






670 


614 


620 


627 


633 


64C 


646 


653 


659 


666 




671 
672 
673 


672 
737 
801 


678 
743 
808 


685 
750 
814 


691 

756 
821 


698 
763 
827 


7C4 
769 
834 


711 
775 
84C 


717 
782 
846 


724 
788 
853 


730 
795 
859 


■1 
.2 
.3 
.4 

• 5 
.6 

• 7 
8 

.9 




1 
1 
2 
3 
3 
4 
5 
5 


% 

3 

9 

9 
5 
2 
8 


674 
675 


866 
930 


872 
937 


879 
943 


885 
949 


892 
956 


898 
962 


904 
969 


911 
975 


917 
982 


924 
988 


676 


994 


*001 


*007 


*014 


*020 


*C27 


*033 


*C39 


*046 


*052 


677 
678 


83 059 
123 


065 

129 


071 
136 


078 

142 


084 
148 


091 
155 


097 
161 


103 
168 


110 
174 


116 
180 


679 


187 
251 


193 
257 


200 


206 


212 


219 


225 


231 


238 


244 


680 


263 


270 


276 


283 


289 


295 


302 


308 




681 


314 


321 


327 


334 


340 


346 


353 


359 


365 


372 




682 


378 


385 


391 


397 


404 


410 


416 


423 


429 


435 






683 


442 


448 


455 


461 


467 


474 


480 


486 


493 


499 






684 


505 


512 


518 


524 


531 


537 


543 


550 


556 


562 






685 


569 


575 


581 


588 


594 


600 


607 


613 


619 


626 






686 


632 


638 


645 


651 


657 


664 


670 


676 


683 


689 




6 


687 


695 


702 


708 


714 


721 


727 


733 


740 


746 


752 


.1 

• 2 

• 3 
.4 
.5 
.6 
.7 
.8 

• 9 


688 
689 


759 
822 


765 
828 


771 
834 


778 
841 


784 
847 

910 


790 
853 


796 
859 


803 
866 


809 
872 


815 
878 


1 

1 

2 

3. 

3- 

4. 

4. 


u 

2 
8 


690 


885 


891 


897 


904 


916 
979 


922 


929 
992 


935 
998 


94l 
*004 


4 



691 


948 


954 


960 


966 


973 


985 


6 
o 


692 
693 


84 010 
073 


017 
079 


023 
086 


029 
092 


035 
.098 


042 
104 


048 
111 


054 
117 


061 
123 


067 
129 


8 


694 


136 


142 


148 


154 


161 


167 


173 


179 


186 


192 


•■* 


695 


198 


204 


211 


217 


223 


229 


236 


242 


248 


254 






696 


261 


267 


273 


279 


286 


292 


298 


304 


311 


317 






697 


323 


329 


335 


342 


348 


354 


360 


367 


373 


379 






698 


385 


392 


398 


404 


410 


416 


423 


429 


435 


441 






699 


447 
510 


454 
516 


460 
522 


466 
528 

3 


472 
534 

4 


479 


485 


491 


497 


503 






700 


541 
5 


547 
6 


553 

7 


559 
8 


565 
9 




N. 





1 


2 


P. 


P. 





534 









TABLE v.— LOGARITHMS OF NIBIBERS. 








N. 





1 
516 


2 

522 


3 


4 


5 

541 
603 




547 
609 


7 

553 
615 


8 
559 


9 

565 


P. P. 


700 


84 510 


528 


534 
596 






701 


572 


578 


584 


590 


62l 


627 




702 


633 


640 


646 


652 


658 


634 


671 


677 


683 


689 






703 


695 


701 


708 


714 


720 


726 


732 


739 


745 


751 






704 


757 


763 


769 


778 


782 


788 


79^ 


8Ct 


806 


813 






705 


819 


825 


831 


837 


843 


849 


856 


862 


868 


874 






706 


880 


883 


893 


899 


905 


911 


917 


923 


929 


936 


t: 




707 


942 


948 


954 


960 


986 


972 


979 


985 


991 


997 


.1 

.2 

• 3 
.4 

• 5 
.6 
.7 
.8 
.9 


0"6 
1-3 
1.9 
2.6 
3.2 
39 
4-5 
5.2 
5.8 




708 


35 003 


009 


015 


021 


023 


034 


040 


046 


052 


058 




709 


064 


070 


077 


083 


089 

150 


095 


101 


107 


113 


119 




710 


126 


132 


138 
199 


144 
205 


156 


162 


168 


174 


181 




711 


187 


193 


211 


217 


223 


229 


236 


242 




712 
713 
714 


248 
309 
370 


254 
315 
376 


230 
321 
382 


286 
327 
388 


272 
333 
394 


278 
339 
400 


284 
345 
406 


290 
351 
412 


297 
357 
418 


303 
363 
424 




715 


430 


433 


443 


449 


455 


461 


467 


473 


479 


485 






716 


491 


497 


503 


509 


515 


521 


527 


533 


540 


546 






717 


552 


558 


534 


570 


573 


582 


588 


594 


600 


606 






718 


612 


618 


624 


630 


636 


642 


648 


655 


661 


667 






719 


673 
733 
793 


679 
739 
799 


685 
745 


691 
751 


697 


703 


709 


715 


721 


727 


6 




720 


757 


763 


769 


775 


781 


787 

847 




721 


805 


811 


817 


823 


829 


835 


841 




722 


853 


859 


865 


872 


878 


884 


890 


896 


902 


908 


• i 
.2 
.3 

• 4 

• 5 

• 6 
.7 

8 
.9 


u 
1 
1 
2 
3 
3 
4 
4 
5 


.o 
2 
8 
4 

6 
2 
8 
4 




723 


914 


920 


928 


932 


933 


944 


950 


956 


962 


968 




724 


974 


980 


936 


992 


998 


^^004 


^010 


*0i6 


=*=022 


^'028 




725 
726 


86 034 
093 


040 
099 


043 
105 


052 
llT 


058 
117 


063 
128 


069 

129 


075 
135 


081 

141 


087 
147 




727 


153 


159 


165 


171 


177 


183 


189 


195 


201 


207 




728 
729 


213 
273 

332 

391 


219 
278 

338 

397 


225 
284 

344 

403 


231 
290 

350 


237 
296 

356 


243 
302 

362 


249 
308 

368 

427 


255 
314 

374 


261 
320 


267 

326 




730 


380 


386 






731 


409 


415 


42l 


433 


439 


445 




732 


451 


457 


483 


489 


475 


481 


486 


492 


498 


504 






733 


510 


516 


522 


528 


534 


54C 


546 


552 


558 


563 






734 


569 


575 


581 


587 


593 


599 


605 


611 


617 


623 






735 


628 


634 


840 


646 


652 


658 


664 


670 


676 


682 






736 


688 


693 


699 


705 


711 


717 


723 


729 


735 


741 


— 




737 


746 


752 


758 


764 


770 


776 


782 


788 


794 


800 


.1 

• 2 
.3 

• 4 
.5 
.6 
.7 
.8 

• 9 


r^''^ 




738 


805 


811 


817 


823 


829 


835 


841 


847 


852 


858 



1 
1 
2 
2 
3 
3 
4 
4 







739 


864 


870 


878 


882 


888 


894 


899 


9C5 


911 


917 


\ 

2 
7 
3 
8 

i 




740 


923 


929 


935 


941 


946 


952 


958 


964 
^•=023 


970 


976 




741 


982 


987 


993 


999 


*005 


*on 


-017 


*028 


*034 




742 
743 
744 


87 040 
099 
157 


046 
104 
163 


052 
110 
169 


058 

116 
175 


06^ 

122 
180 


069 
128 
186 


075 
134 
192 


081 

140 
198 


087 
145 
204 


093 

151 
210 




745 


215 


221 


227 


233 


239 


245 


250 


256 


262 


268 






746 


274 


279 


285 


291 


297 


303 


309 


314 


320 


326 






747 


332 


338 


343 


349 


355 


361 


367 


372 


378 


384 






748 


390 


396 


402 


407 


413 


419 


425 


431 


436 


449 






749 


448 


454 


460 


465 


471 


477 


483 


489 


494 


500 
558 

9 






750 


506 


512 

1 


517 
2 


523 
3 


529 
4 


535 
5 


541 
6 


546 

7 


552 
8 




N. 





P. P. 





535 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 


2 

517 


3 

523 


4 

529 
587 


5 

535 
593 


6 

541 


7 
546 


8 
552 


9 


750 


87 506 
564 


512 


558 


751 


570 


575 


581 


598 


604 


610 


616 


752 


622 


627 


633 


639 


645 


650 


656 


662 


668 


673 


753 


679 


6851 691 


697 


i 702 


708 714 


720 


725 


731 


754 


737 


^ 743 i 748 


754 


1 760 


766 771 


777 


783 


789 


755 


794 


800 806 


812 


817 


823 829 


835 


840 


846 


756 


852 


858 


863 


869 


' 875 


881' 886 


892 


898 


904 


757 


909 


915 


921 


927 


, 932 


938 944 


949 


955 


961 


758 


967 


972 


978 


984 


990 


995 *001 


j*007 


*012!*018 


759 


88 024 


030 


035 


041 


047 


053 058 


064 


070 


075 


760 


081 


087 


093 


098 


104 


no! 115 


121 


127 


133 


761 


138 


144 


150 


155 


1 161 


167 172 


178 


184 


190 


762 


195 


1 201 


207 


2lS 


218 


224 229 


235 


241 


247 


763 


252 


1 258 


264 


269 


275 


28l! 286 


292 


298 


303 


764 


309 


' 315 


320 


326 


332 


337 343 


349 


355 


360 


765 


366 


372 


377 


383 


389 


394 400 


406 


411 


417 


766 


423 


428 


434 


440 


445 


451' 457 


462 


468 


474 


767 


479 


485 


491 


496 


502 


508 513 


519 


525 


530 


768 


536 


542 


547 


553 


558 


564 570 


575 


58] 


587 


769 


595 


598 


604 


609 


615 


621; 626 


632 


638 


643 


770 


649 


654 


660 


666 


671 


677] 683 


688 


694 


700 


771 


705 


711 


716 


722 


728 


733i 739 


745 


750 


756 


772 


761 


767 


773 


778 


784 


790, 795 


801 


806 


812 


773 


818 


8231 829 


835 


840 


846, 851 


857 


863 


868 


774 


874 


8791 885 


891 


896 


902 907 


913 


919 


924 


775 


930 


9361 941 


947 


952 


958 964 


969 


975 


980 


776 


986 


992 997 


*003 


'^OOS 


="014*039 


*025 


*031 


*036 


777 


89 042 


047i 053 


059 


.064 


070 075 


081 


087 


092 


778 


098 


103 


109 


114 


120 


126 131 


137 


142 


148 


779 


153 


159 


165 


170 


176 


181: 187 


193 


198 


204 


780 


209 


215 


220 


226 


231 


237 243 


248 


254 


259 


781 


265 


270 


276 


282 


287 


293 298 


304 


309 


315 


782 


320 


326 


332 


337 


343 


348 354 


359 


365 


370 


783 


376 


381 


387 


393 


398 


404 409 


415 


420 


426 


784 


431 


437 442 


448 


454 


459 465 


470 


476 


481 


785 


487 


492 


498 


503 


509 


514 520 


525 


531 


536 


786 


542 


548 


553 


559 


564 


570 575 


581 


586 


592 


787 


597 


603 608 


614 


619 


625 630 


636 


641 


647 


788 


652 


658 663 


669 


674 


680 685 


691 


696 


702 


789 


707 


713 


718 


724 


729 


735 740 


746 


751 


757 


790 


762 


768 


773 


779 


784 


790 795 


801 


806 


812 


791 


817 


823 828 


834 


839 


845 850 


856 


861 


867 


792 


872 


878 883 


889 


894 


900 905 


911 


916 


922 


793 


927 


933! 938 


943 


949 


954 960 


965 


971 


976 


794 


982 


9871 993| 


998 


*004 


*009 *015 


*020 


*026 


*031 


795 


90 036i 


042 0471 


053 


058 


064 069 


075 


080 


08P 


796 


091 


097 


102: 


1071 


113 


118 124 


129 


135 


140 


797 


146 


151 


156; 


1621 


167 


173 178 


184 


189 


195 


798 


200 


205 


911' 


216i 


222 


227 233 


238 


244 


249 


799 


254 


260 


265; 


2711 


276 


282 287| 




298 


303 


800 


309 


314 

1 


320! 
2 


325 1 
3 


330 


336, 341 


347 
7 


352 
8 


358 
9 


N, 





4 


' 


6 



P. p. 



1 


0. 


2 


1. 


3 


1. 


4 


2. 


5 


3. 


6 


3- 


7 


4. 


8 


4- 


9 


5. 



1 





.5 


2 


1 


1 


3 


1 


6 


4 


2 


2 


5 


2 


7 


6 


3 


3 


7 


3 


8 


8 


4 


4 


9 


4 


9 



5 

0-5 
1-0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4-5 




I\ P. 



536 



TABLE v.— LOGARITHMS OF NUxMBERS. 



N. 





1 


2 


3 


4 

330 


5 

336 


6 

34l 


7 


8 


9 

358 


P. P. 


800 


90 309 


314 


320 


325 


347 


352 




801 


363 


368 


374 


379 


385 


390 


396 


401 


406 


412 




802 


417 


423 


428 


433 


439 


444 


450 


455 


460 


466 




803 


471 


477 


482 


488 


493 


498 


504 


509 


515 


520 




804 


525 


531 


536 


542 


547 


552 


558 


563 


569 


574 




805 


579 


585 


590 


596 


601 


606 


612 


617 


622 


628 




806 


633 


639 


644 


649 


655 


660 


666 


671 


676 


682 




807 


687 


692 


698 


703 


709 


714 


719 


725 


730 


736 




808 


741 


746 


752 


757 


762 


768 


773 


778 


784 


789 




809 


795 

848 

902 
955 


800 


805 


811 


816 


821 


827 


832 


838 


843 




810 


854 


859 


864 


870 


875 


£80 


886 

939 
993 


891 

945 
998 


896 

95^ 
=^003 




811 
812 


907 
961 


913 
966 


918 
971 


923 
977 


929 
982 


934 
987 




•1 
.2 
.3 
.4 
.5 
.6 
.7 
8 
.9 


0^5 

1 1 


813 
814 
815 
813 
817 
818 
819 


91 009 
062 
116 
169 
222 
275 
328 


014 
068 
121 
174 
227 
280 
333 


019 
073 
126 
179 
233 
286 
339 


025 
078 
131 
185 
238 
291 
344 


030 
084 
137 
190 
243 
296 
349 


036 
089 
142 
195 
249 
302 
355 


04] 
094 
147 
201 
254 
307 
360 


046 
100 
153 
206 
259 
312 
365 


052 
105 
158 
211 
264 
318 
371 


057 
110 
163 
217 
270 
323 
376 




1 

2 
2 
3 
3 
4 
4 


• X 

.6 
-2 
■7 


820 


38l 


386 


392 


397 


402 


408 


413 


418 


423 


429 




821 


434 


439 


445 


450 


455 


461 


466 


471 


476 


482 




822 


487 


492 


497 


503 


508 


513 


519 


524 


529 


534 




823 


540 


545 


550 


556 


561 


566 


571 


577 


582 


587 




824 


592 


598 


603 


608 


614 


619 


624 


629 


635 


640 




825 


645 


650 


656 


661 


666 


671 


677 


682 


687 


692 




826 


698 


703 


708 


714 


719 


724 


729 


735 


740 


745 




827 


750 


756 


761 


766 


771 


777 


782 


787 


792 


798 




828 


803 


808 


813 


819 


824 


829 


834 


839 


845 


850 




829 


855 


860 


866 


871 


876 


881 


887 


892 
944 


897 
949 


902 
955 




830 


908 


913 


918 


923 


928 


934 


939 




831 
832 


960 
92 012 


965 
017 


970 
023 


976 
028 


981 
033 


986 
038 


991 
0';:3 


996 
049 


*002 
054 


*007 
059 




1 
2 
3 
4 
5 
6 
7 
8 
9 


«. 

1 
1 
2 
2 
3 
3 
4 
4 


> 

5 


n 


833 


064 


069 


075 


080 


085 


090 


096 


101 


106 


111 




834 


116 


122 


127 


132 


137 


142 


148 


153 


158 


163 




5 


5 

5 

5 


835 
836 
837 
838 
839 


168 
220 
272 
324 
376 

428 


174 
226 
277 
329 
381 

433 


179 
231 
283 
335 
386 

438 


184 
236 
288 
34C 
391 


189 
241 
293 
345 
397 


194 
246 
298 
350 
402 


200 
252 
303 
355 

407 

459 


205 
257 
309 
360 
412 


210 
262 
314 
366 
417 

469 


215 
267 
319 
371 
423 

474 




840 


443 


448 


454 


464 




841 


479 


485 


490 


495 


500 


505 


510 


515 


521 


526 




842 


531 


536 


541 


546 


552 


557 


562 


567 


572 


577 




843 


583 


588 


593 


598 


603 


608 


613 


619 


624 


629 




844 


634 


639 


644 


649 


655 


660 


665 


670 


675 


680 




845 


685 


691 


696 


701 


706 


711 


716 


721 


727 


732 




846 


737 


742 


747 


752 


757 


762 


768 


773 


778 


783 




847 


788 


793 


798 


803 


809 


814 


819 


824 


829 


834 




848 


839 


844 


850 


855 


860 


865 


870 


875 


880 


885 




849 


891 


896 


901 


906 


911 


916 


921 


926 


931 


937 




850 


942 


947 
1 


952 
2 


957 
3 


962 


967 
5 


972 


977 
7 


982 
8 


988 
9 




N. 





4 


6 


P. P. 



537 



TABLE V— LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 

962 


5 

967 


6 

972 


7 
977 


8 
982 


9 

988 


P. P. 


850 


92 942 


947 


952 


957 




851 
852 
853 
854 
855 
856 
857 
858 
859 


993 
93 044 
095 
146 
196 
247 
298 
348 
399 


998 
049 
100 
151 
201 
252 
303 
354 
404 


*003 
054 
105 
156 
207 
257 
308 
359 
409 


*008 
059 
110 
161 
212 
262 
313 
364 
414 


*013 
064 
115 
166 
217 
267 
318 
369 
419 


*018 
069 
120 
171 
222 
272 
323 
374 
424 


*023 
074 
125 
176 
227 
278 
328 
379 
429 


*028 
079 
130 
181 
232 
283 
333 
384 
434 


*034 
084 
135 
186 
237 
288 
338 
389 
439 


*039 
090 
140 
191 
242 
293 
343 
394 
445 




•1 

.2 

3 

.4 

• 5 

• 6 
•7 

8 
9 




1 
1 
2 
2 
3 
3 
4 
4 


5 

5 

-.1 


860 


450 


455 


460 


465 


470 


475 


480 


485 

535 
586 
636 
686 
736 
787 
837 
887 
937 


490 

540 
591 
641 
691 
742 
792 
842 
892 
942 


495 

545 
596 
646 
696 
747 
797 
847 
897 
947 


• 2 

• 7 


861 
862 
863 
864 
865 
866 
867 
868 
869 


500 
550 
601 
651 
701 
752 
802 
852 
902 


505 
556 
606 
656 
706 
757 
807 
857 
907 


510 
561 
611 
661 
711 
762 
812 
862 
912 


515 
566 
616 
666 
716 
767 
817 
867 
917 


520 
571 
621 
671 
721 
772 
822 
872 
922 


525 
576 
626 
676 
726 
777 
827 
877 
927 


530 
581 
631 
681 
731 
782 
832 
882 
932 


3 

• 8 
•4 

• 9 


870 


952 


957 


962 


967 


972 


977 


982 


987 


992 


997 




871 
872 
873 
874 
875 
876 
877 
878 
879 


94 002 
05l 
lOl 
151 
201 
250 
300 
349 
399 


007 
056 
106 
156 
208 
255 
305 
354 
404 


012 
061 
111 
161 
210 
260 
310 
359 
409 


017 
066 
116 
166 
215 
265 
315 
364 
413 


022 
071 
121 
171 
220 
270 
320 
369 
418 


026 
076 
126 
176 
225 
275 
324 
374 
423 


03l 
08l 
13l 
181 
230 
280 
329 
379 
428 


036 
086 
136 
186 
235 
285 
334 
384 
433 


04l 
091 
141 
191 
240 
290 
339 
389 
438 


046 
096 
146 
196 
245 
295 
344 
394 
443 




-1 
.2 
3 
4 
5 
6 
7 
8 
9 




1 
1 
2 
2 
3 
3 

4. 
4 


5 

5 

5 

5 

5 

5 


880 


448 


453 


458 


463 


468 


473 


478 


483 


487 


492 




881 
882 
883 
884 
885 
886 
887 
888 
889 


497 
547 
596 
645 
694 
743 
792 
841 
890 

939 


502 
552 
601 
650 
699 
748 
797 
846 
895 


507 
556 
606 
655 
704 
753 
802 
851 
900 


512 
561 
611 
660 
709 
758 
807 
858 
905 


517 
566 
615 
665 
714 
763 
812 
861 
909 


522 
571 
620 
670 
719 
768 
817 
865 
914 


527 
576 
625 
674 
724 
773 
821 
870 
919 


532 
581 
630 
679 
728 
777 
826 
875 
924 


537 
586 
635 
684 
733 
782 
831 
880 
929 

978 


542 
591 
640 
689 
738 
787 
836 
885 
934 

983 




1 
2 
3 
4 
5 
6 
7 
8 
9 


0. 



1- 

1. 

2- 

2- 

3 

3. 

4- 


il 


890 


944 


949 


953 


958 


963 


968 


973 




891 
892 
893 
894 
895 
896 
897 
898 
899 


988 

95 036 
085 
134 
182 
231 
279 
327 
376 

424 


992 
041 
090 
138 
187 
235 
284 
332 
381 

429 

1 


997 
046 
095 
143 
192 
240 
289 
337 
385 


*002 
051 
099 
148 
197 
245 
294 
342 
390 


*007 
058 
104 
153 
201 
250 
298 
347 
395 


*012 
061 
109 
158 
206 
255 
303 
352 
400 


*017 
065 
114 
163 
211 
260 
308 
356 
405 


*022 
070 
119 
167 
216 
264 
313 
361 
410 


=^026 
075 
124 
172 
221 
269 
318 
366 
414 


03l 
080 
129 
177 
226 
274 
323 
371 
419 


7 
1 
6 



900 


434 
2 


438 


443 


448 
5 


453 


458 


463 
8 


467 
9 




N. 





3 


4 


6 


7 


P.P. 






538 







TABLE V. 


—LOGARITHMS OF NUMBERS. 








N. 





1 


2 

434 


3 

438 


4 

443 


5 

448 


6 

453 


7 
458 


8 
463 


9 

467 


P. P. 


^00 


95 424 


429 



































901 


472 


477 


482 


487 


492 


496 


501 


506 


511 


516 




902 


520 


525 


530 


535 


540 


544 


549 


554 


bb9 


564 




903 


569 


573 


578 


583 


588 


593 


597 


602 


6CV 


612 




904 


617 


621 


626 


631 


636 


641 


64b 


6b0 


bbb 


660 




905 


665 


669 


674 


679 


684 


689 


693 


698 


VU3 


708 




906 


713 


717 


722 


727 


732 


737 


741 


74b 


7bl 


7b6 




907 


760 


765 


770 


775 


780 


784 


789 


794 


799 


804 




908 


808 


813 


818 


823 


827 


832 


837 


842 


847 


8bi 




909 


856 


861 


866 


870 


875 


880 


885 


890 


894 


899 




910 


904 
952 


909 


913 


918 


923 


928 


933 


937 


942 


947 




911 


956 


96l 


966 


971 


975 


980 


985 


990 


994 


.1 


5 

O*^ 


912 


999 


*004 


*009 


*014 


*018 


*023 


*028 


*033 


-^037 


^042 




2 


1 





913 


96 047 


052 


056 


061 


066 


071 


075 


080 


08b 


090 




3 


1 


5 


914 


094 


099 


104 


109 


113 


118 


123 


128 


132 


13V 




4 


2 





915 


142 


147 


151 


156 


161 


166 


170 


IVb 


180 


18b 




5 


2 


5 


916 


189 


194 


199 


204 


208 


213 


218 


222 


227 


232 




6 


3 





917 


237 


241 


246 


251 


256 


260 


265 


27U 


27b 


279 




7 


3 


5 


918 


284 


289 


293 


298 


303 


308 


312 


317 


322 


327 




n 


4 





919 


331 
379 
426 


336 


341 


345 


350 


355 


360 


364 


369 


374 




9 


4 


5 


930 


383 
430 


388 


393 


397 


402 


407 


412 


416 


421 




921 


435 


440 


445 


449 


454 


459 


463 


468 




922 


473 


478 


482 


487 


492 


496 


501 


506 


511 


bib 




923 


520 


525 


529 


534 


539 


543 


548 


553 


558 


562 




924 


567 


572 


576 


581 


586 


590 


595 


600 


605 


609 




925 


614 


619 


623 


628 


633 


637 


642 


647 


651 


6bb 




926 


661 


666 


670 


675 


680 


684 


689 


694 


698 


703 




927 


708 


712 


717 


722 


726 


731 


736 


741 


745 


750 




928 


755 


759 


764 


769 


773 


778 


783 


787 


792 


797 




929 


801 


806 


811 


815 


820 


825 


829 


834 


839 


843 




930 


848 


853 


857 


862 


867 


871 


876 


881 


885 


890 




931 


895 


899 


904 


909 


913 918 


923 


927 


932 


937 


.1 


0*2 


932 


941 


946 


951 


955 


960 


965 


969 


974 


979 


983 




2 





9 


933 


988 


993 


997 


*002 


*007 


*011 


*016 


*020 


=^^025 


*030 




1 


3 


934 


97 034 


039 


044 


048 


053 


058 


062 


067 


072 


076 




4 


1 




935 


081 


086 


090 


095 


099 


104 


109 


113 


118 


123 




5 


2 


2 


936 


127 


132 


137 


141 


146 


151 


155 


160 


164 


169 




6 


? 


7 


937 


174 


178 


183 


188 


192 


197 


202 


20b 


211 


2ib 




7 
8 


3 


I 


938 


220 


225 


225 


234 


239 


243 


248 


252 


257 


262 




3 




939 


266 


271 


276 


280 


285 


289 


294 


299 


303 


308 




.9 


4 


Q 


940 


313 


317 


322 


326 


331 


336 


340 


345 


349 


354 




941 


359 


363 


368 


373 


377 


382 


386 


39l 


396 


400 




942 


405 


409 


414 


419 


423 


428 


432 


437 


442 


44b 




943 


451 


456 


460 


465 


469 


474 


479 


483 


488 


492 




944 


497 


502 


506 


511 


515 


520 


525 


529 


534 


538 




945 


543 


548 


552 


557 


561 


566 


570 


575 


580 


584 




946 


589 


593 


598 


603 


607 


612 


616 


621 


626 


630 




947 


635 


639 


644 


649 


653 


658 


662 


667 


671 


67b 




948 


681 


685 


690 


694 


699 


703 


708 


713 


717 


722 




949 


726 


731 


736 


740 


745 


749 


754 


758 


763 


768 
813 

9 




950 


772 


777 
1 


781 
2 


786 


790 


795 
5 


800 
6 


804 

7 


809 
8 




N. 





3 


4 


P. P, 



539 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 
777 


2 

78l 


3 

786 


4 


5 


6 


7 


8 
809 


9 

813 


P. 


P. 


950 


97 772 
818 


790 


795 


800 


804 






951 


822 


827 


831 


836 


841 


845 


850 


854 


859 


952 


863 


868 


873 


877 


882 


886 


891 


895 


900 


904 






953 


909 


914 


918 


923 


927 


932 


936 


941 


945 


950 






954 


955 


959 


984 


968 


973 


977 


982 


986 


991 


996 






955 


98 000 


005 


009 


014 


018 


023 


027 


032 


036 


041 






956 


046 


050 


055 


059 


064 


068 


073 


077 


082 


086 




5 

0.5 
1.0 

1.5 
2.0 
2-5 
30 
35 
4-0 
4.5 


957 
958 
959 


091 
136 
182 


095 
141 
186 


100 
145 
191 


105 
150 
195 


109 
154 
200 


114 
159 
204 


118 

163 
209 


123 
168 
213 


127 
173 
218 


132 
177 
222 


.1 

:i 

.4 
.5 
.6 
• 7 
.8 
.9 


960 


227 


231 


236 


240 


245 


249 


254 


259 


263 


268 


931 
962 
983 
964 


272 
317 
362 
407 


277 
322 
367 
412 


28l 
326 
371 
416 


286 
331 
376 
421 


290 
335 
380 

425 


295 
340 
385 
430 


299 
344 
389 

434 


304 
349 
394 
439 


308 
353 
398 

443 


313 
358 
403 
448 


965 


452 


457 


461 


466 


470 


475 


479 


484 


488 


493 






986 


497 


502 


506 


511 


515 


520 


524 


529 


533 


538 






937 


542 


547 


551 


556 


560 


565 


569 


574 


578 


583 






968 


587 


592 


596 


601 


605 


610 


614 


619 


623 


628 






969 


632 


637 


641 


646 


650 


655 


659 


663 


868 


672 


.1 
.2 
-3 
.4 
-5 
.6 
.7 
.8 
.9 




970 


677 


681 


686 


690 


695 


699 


704 


708 


713 


717 


971 
972 
973 
974 
975 
976 


722 
766 
811 
856 
900 
945 


726 
771 
815 
860 
905 
949 


731 
775 
820 
865 
909 
954 


735 
780 
824 
869 
914 
958 


740 
784 
829 
873 
918 
963 


744 
789 
833 
878 
922 
967 


749 
793 
838 
882 
927 
971 


753 
798 
842 
887 
931 
976 


757 
802 
847 
891 
936 
980 


762 
807 
851 
896 
940 
985 




1 
1 
2 
2 
3 
3 
4 


9 
3 
8 
2 
7 
I 
6 



977 
978 
979 


989 

99 034 

078 


994 
038 
082 


998 
043 
087 


*003 
047 
091 

136 


*007 
051 
098 

140 


*011 
058 
100 


*016 
060 
105 


*020 
065 
109 


*025 
089 
113 


*029 
074 
118 


980 


122 


127 


131 


145 
189 


149 


153 


158 


162 
206 




981 


167 


171 


176 


180 


184 


193 


198 


202 




932 


211 


215 


220 


224 


229 


233 


237 


242 


246 


251 






983 


255 


260 


264 


268 


273 


277 


282 


286 


290 


295 






984 


299 


304 


308 


312 


317 


321 


326 


330 


335 


339 






985 


343 


348 


352 


357 


361 


385 


370 


374 


379 


383 






988 


387 


392 


396 


401 


405 


409 


414 


418 


423 


427 




4 

0.4 
0.8 

1-2 


987 


431 


436 


440 


445 


449 


453 


458 


462 


467 


471 


.1 
.2 

• 3 
.4 
.5 

• 6 
.7 
.8 
.9 


988 
989 


475 
519 


480 
524 


484 
528 


489 
533 


493 
537 


497 
541 


502 
548 


506 
550 


511 
554 


515 
559 


990 


563 


568 


572 


576 


581 


585 


590 


594 


598 


803 


l.B 
2.0 


991 
992 
993 
994 


607 
651 
695 
738 


611 
655 
699 
743 


616 
660 
703 
747 


620 
664 
708 
75T 


625 
668 
712 
756 


829 
673 
717 
780 


833 
877 
721 
785 


838 
882 
725 
789 


642 
886 
730 
773 


847 
690 
734 
778 


2.4 
2.8 
3.2 
3.6 


995 


782 


786 


791 


795 


800 


804 


808 


813 


817 


821 






996 


826 


830 


834 


839 


843 


847 


S59. 


856 


861 


885 






997 


869 


874 


878 


882 


887 


891 


895 


900 


904 


908 






998 


913 


917 


922 


926 


930 


935 


939 


943 


94R 


95? 






999 


956 


961 


965 


969 


974 


978 


982 


987 


991 


995 




i 


1000 


000 000 


004 

1 


008 
2 


013 
3 


017 
4 


021 
5 


028 
6 


030 

7 


034 
8 


039 


_ 1 


N. 





P. ] 


P. 


■^ 



540 







TABLE V.- 


-LOGARITHMS OF NUMBEBS. 










N. 





1 
043 


2 

087 


3 

130 


4 

173 


5 

217 


6 

260 


7 
304 


8 
347 


9 

39C 


P. P. 


\1000 

01 


000 000 




434 


477 


521 


564 


607 


651 


694 


737^ 78li 824 




02 


867 


911 


954 


997 


1*041 


*084 


*127 


*171 *214 *257 




03 


001 301 


344 


387 


431 


, 474 


517 


560 


604 647 690 




04 


733 


777 


820 


863 


' 906 


950 


993 


*036 *079 *123 




05 


002 166 


209 


252 


295 


339 


382 


425 


468 511 


555 




06 
07 
08 
09 


598 

003 029 

460 

891 


641 
072 
503 
934 


684 
115 
546 
977 


727 

159 

590 

*020 


770 

202 

633 

*063 


814 

245 

676 

*106 


857 
288 

719 
*149 


900 943 

331 374 

762 805 

*192 *235 


986 

417 

848 

*278 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


43 

4.3 
87 

130 
17.4 
21.7 
26-1 


4: 

4 
8 

12 
17 
21 
25 
30 
34 


5 

3 
6 
9 


1010 


004 321 


364 


407 


450 


493 


536 


579 


622 665 


708 


2 
5 


11 


751 


794 


837 


880 


923 


966 


*009 


*05l *094 


*137 


8 

1 
4 


12 
13 


005 180 
609 


223 
652 


266 
695 


309 
738 


352 
781 


395 
824 


438 
866 


481 523 
909 952 


566 
995 


34 


8 


14 


006 038 


081 


123 


166 


209 


252 


295 


337 380 


423 


OS J. 


OO • 1 


15 


466 


509 


551 


594 


637 


680 


722 


765 808 


851 




16 


893 


936 


979 


*022 *064| 


*107 


*15G 


*193 *235 


*278 




17 


007 321 


363 


406 


449 


491 


534 


577 


620' 662 


705 




18 


748 


790 


833 


875 


918 


961 


*C03 


*046*089 


131 




19 


008 174 


217 


259 


302 


344 


387 


430 


472 515 


557 




1030 


600 


642 


685 


728 


770 


813 


855 


898 940 


983 




21 


009 025 


068 


111 


153 


196 


238 


281 


323 366 


408 


42 42 

1 A n\ A rt 


22 


451 


493 


536 


578 


621 


663 


7C6 


748 790 


833 


.2 
.3 
.4 
.5 
• 6 
.7 
.8 
.9 


8 

12 
17 
21 
25 
29 

38 


C 
2 
5 
7 

2 


8 

12 
16 
21 
25 
29 
33 
37 


A 


23 
24 


875 
010 300 


918 
342 


960 
385 


*003 *045 
427 469 


*088 
512 


*13C 
554 


*172 *215 
596 639 


*257 
681 


6 
8 

2 
4 
6 
8 


25 
26 
27 


724 

Oil 147 

570 


766 
189 
612 


808 
232 
655 


851 
274 
697 


893 
316 
739 


935 
359 
782 


978 
401 

824 


*020 *062 
443 486 
866 908 


*105 
528 
951 


28 


9931*035 


*077 


'*=120|*162 


*204 


*246 


*288 *331 


*373 


29 


012 415 457 


500 


542j 584 


626 


668 


710 753 


795 


1030 


837 879 


921 


963 *006 


*048 


*090 


*132 174 


216 




31 


013 258 


301 


343 


385 


427 


469 


511 


553 595 


637 




32 


679 


722 


764 


806 


848 


890 


932 


974*016 


*058 




33 


014 100 


142 


184 


226 


268 


310, 


352 


394 436 


478 




34 


520 562! 


604 


646; 688 


730 


7/2 


814 856 


898 




35 


940 


982 


*024 


*066*108 


*150 


*192 


*234*276 


*318 




36 


015 360 


401 


443 


485 527 


569 


611 


653 695 


737 


aY a -i 


37 


779 


820 


862 


904; 946 


988 


*030 


*072*113 


155 


.1 
.2 

• 3 

• 4 
.5 
.6 
.7 
.8 

q 


1 =r 


4. 1 


38 


016 197 


239 


281 


323^ 364 


406 


448 


490 532 


573 


8 

12 
16 
20 
24 
29 
33 

•31 


3 
4 
6 
7 
9 

2 

Q 


8 

12 
16 
20 
24 
28 
32 
9ft 


2 
3 


39 


615 


657 


699 


741 i 782 


824 
242 


866 


908 950 


991 


1040 


017 033 


075 


117 


158 


200 


284 


325 367 


409 


4 
5 


41 


450 


492 


534 


576 


617 


659 


701 


742 784 


826 


6 
7 
8 

Q 


42 


867 


909 


951 


992*034 


*076 


*117 


*159 *201 


■^242 


43 


018 284 


326 


387 


409^ 451 


492. 


534 


575 617 


659 


44 


700 


742 


783 


825 


867 


908 


950 


991 *033 


^074 




45 


019 116 


158 


199 


241 


282 


324 


365 


4C7 448 


490 




46 


531 


573 


614 


656 


697 


739 


780 


822 863 


905 




47 


946 


988 


*029 


*071 


*112 


*154 


*195 


*237 ^278 


=^320 




48 


020 361 


402 


444 


485 


527 


568 


61C 


651 692 


734 




49 


775 
021 189 


817 


858 


899 


941 


982 


*024 


*065*106 


=^148 




1050 


230 

1 


272 
2 


313 
3 


354 
4 


396 
5 


437 
6 


478 520 


56l 




N. 





7 


8 


9 


P. P. 



541 









TABLJ 


I V.- 


-LOGARITHMS OF NUMBERS. 






N. 





1 


2 

272 


3 


4 


5 


6 


7 
478 


8 
520 


9 

561 


P 


.P. 


1050 


021 


189 


230 


313 


354 


396 


437 


•1 


41 

4.1 


51 


602 


644 


685 


726 


768 


809 


850 


892 


933 


974 


52 


022 


015 


057 


098 


139 


181 


222 


263 


304 


346 


387 


• 2 


8 


3 


53 




428 


469 


511 


552 


593 


634 


676 


717 


758 


799 


3 


]2 


4 


54 




840 


882 


923 


964 


*005 


*046 


*088 


*129 


*170 


*211 


• 4 


16 


6 


55 


023 


252 


293 


335 


376 


417 


458 


499 


540 


581 


623 


• 5 


20 


7 


56 




664 


705 


746 


787 


828 


869 


910 


951 


993 


*034 


• 6 


24 




57 


024 


075 


116 


157 


198 


239 


280 


321 


362 


403 


444 


• 7 


29 





58 




485 


526 


568 


609 


650 


691 


732 


773 


814 


855 


8 


33 


2 


59 


025 


896 


937 


978 


^019 


*060 


*101 


*142 


*183 


=^224 


*265 


9 
• 1 


37 


3 


1060 


306 


347 


388 


429 


469 


510 


551 


592 


633 


674 


41 

4-1 


61 


715 


756 


797 


838 


879 


920 


961 


*002 


*042 


*083 


62 


02-6 


124 


165 


206 


247 


288 


329 


370 


410 


451 


492 


.2 


8 


2 


63 




533 


574 


615 


656 


696 


737 


778 


819 


860 


901 


• 3 


12 


3 


64 




941 


982 


*023 


*064 


*105 


*145 


n86 


*227 


*268 


*309 


.4. 


16 


4 


65 


027 


349 


390 


431 


472 


512 


553 


594 


635 


675 


716 


• 5 


20 


5 


66 




757 


798 


838 


879 


920 


961 


*001 


^042 


*083 


*123 


• 6 


24 


6 


67 


028 


164 


205 


246 


286 


327 


368 


408 


449 


490 


530 


•7 


28 


7 


68 




571 


612 


652 


693 


734 


774 


815 


856 


896 


937 


8 


32 


8 


69 


029 


977 


*018 


*059 


*099 


*140 


*181 


*22] 


*262 


*302 


*343 


9 

•1 


36 


9 


1070 


384 


424 


465 


505 


546 


586 


627 


668 


708 


749 


40 

4-0 


71 


789 


830 


870 


911 


95l 


99.2 


*032 


*073 


*114 


*154 


72 


030 


195 


235 


276 


316 


357 


397 


438 


478 


519 


559 


• 2 


8 


1 


73 




599 


640 


680 


721 


781 


802 


842 


883 


923 


964 


3 


12 


1 


74 


031 


004 


044 


085 


125 


166 


206 


247 


287 


327 


368 


•4 


16 


2 


75 




408 


449 


489 


529 


570 


610 


651 


691 


731 


772 


.5 


20 


2 


76 




812 


852 


893 


933 


973 


*014 


*054 


*094 


*135 


*175 


• 6 


24 


3 


77 


032 


215 


256 


296 


336 


377 


417 


457 


498 


538 


578 


•7 


28 


3 


78 




619 


659 


699 


739 


780 


820 


860 


900 


941 


98] 


8 


32 


4 


79 


033 


021 


081 


102 


142 


182 


222 


263 


303 


343 


383 


9 
.1 


36 


4 


1080 


424 


464 


504 


544 


584 


625 


665 


705 


745 


785 




81 


825 


866 


906 


946 


986 


*026 


*0S6 


*107 


147 


187 


40 

40 


82 


034 


227 


267 


307 


347 


388 


428 


438 


508 


548 


588 


.2 


8 





83 




628 


668 


708 


748 


789 829 


839 


909 


949 


989 


• 3 


12 





84 


035 


029 


069 


109 


149 


189 


229 


289 


309 


349 


389 


.4 


16 





85 




429 


470 


510 


550 


590 


630 


670 


710 


750 


790 


.5 


20 





86 




830 


870 


910 


950 


990 


=*=029 


*069 


-109 


*149 


*189 


• 6 


24 





87 


036 


229 


269 


309 


349 


389 


429 


469 


509 


549 


589 


.7 


28 





88 




629 


669 


708 


748 


783 


828 


868 


908 


948 


988 


8 


32 





89 


037 


028 


068 


107 


147 


187 


227 


267 


307 


347 


386 


9 

.] 


36 





1090 


426 


466 


506 


546 


586 


625 


665 


705 


745 


785 


39 

39 


91 


825 


864 


904 


944 


984 


=^023 


*063 


*103 


143 


183 


92 


038 


222 


262 


302 


342 


381 


421 


461 


501 


540 


580 


.2 


7 


9 


93 




620 


660 


699 


739 


779 


819 


' 858 


898 


938 


977 


• 3 


11 


8 


94 


039 


017 


057 


096 


136 


176 


216 


255 


295 


335 


374 


• 4 


15 


8 


95 




414 


454 


493 


533 


572 


612 


652 


691 


731 


771 


.5 


19 


7 


96 




810 


850 


890 


929 


969 


*008 


*048 


*088 


*127 


*167 


.6 


23 


■Z 


97 


040 


206 


246 


286 


325 


365 


404 


444 


483 


523 


563 


• 7 


27 


• 6 


98 




602 


642 


681 


721 


760 


800 


839 


879 


918 


958 


8 


31 


6 


99 


041 


997 


*037 


'•=076 


*116 


*155 


*195 


*234 


*274 


*313 


*353 


• 9 


35 


• 5 


1100 


392 


432 


471 
2 


511 


550 


590 
5 


629 
6 


669 

7 


708 
8 


748 
9 




N. 


( 


[) 


1 


3 


4 


P 


. P. 



542 



TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. 



Log sin (f, = log <l>'' -}- S. 




0° 


log<^' 


' = log 


sin (^ + *S' 


Log tan cf> = log <f>'' + T. 




log 0' 


' = log 


tan + T" 


n 


/ 


S 


T L< 


>g. Sin. 


S' 


T' 


I.og. Tan. 








4.685 57 


57 


CO 


5.314 42 


42 


00 


60 


1 


57 


57 6 


.46 372 


42 


42 


6-46 372 


120 


2 


57 


57 


.76 475 


42 


42 


• 76 475 


180 


3 


57 


57 


• 94 084 


42 


42 


• 94 084 


240 


4 
5 


57 


57 7 


-06 578 


42 
5.314 42 


4? 


1 7^06 578 


300 


4.685 57 


57 7 


-16 269 


42 


i 7 16 269 


360 


6 


57 


57 


-24 187 


42 


42 


1 -24 188 


420 


7 


57 


57 


-30 882 


42 


42 


1 -30 882 


480 


8 


57 


57 


36 681 


42 


42 


j 36 681 


540 


9 


57 
4.68^ 57 


57 


• 41 797 


42 


42 
42 


1 .41 797 


600 


10 


57 7 


• 48 372 


5.314 42 


< V 46 372 


660 


11 


57 


-50 512 


42 


42 


• 50 512 


720 


12 


57 


57 


-54 290 


42 


42 


• 54 291 


780 


13 


57 


57 


57 767 


42 


42 


1 57 767 


840 


14 


57 


57 


60 985 


42 


42 


' •60 985 


900 


15 


4-685 57 


58 7 


63 98l 


5.314 42 


42 


7 63 982 


960 


16 


57 


58 


-66 784 


42 


42 


• 66 785 


1020 


17 


57 


58 


-69 417 ■ 


45 


42 


-69 418 


1080 


18 


57 


58 


-71 899 


42 


42 


-71 900 


1140 


19 


57 


58 


- 74 248 


42 


4? 
42 


- 74 248 


1200 


30 


4.685 57 


58 7 


• 76 475 


5.314 43 


i 7-76 476 


1260 


21 


57 


58 


-78 594 


43 


42 


1 -78 595 


1320 


22 


57 


58 


-80 614 


43 


42 


-80 615 


1380 


23 


57 


58 


-82 545 


43 


42 


• 82 546 


1440 


24 


57 


58 


• 84 393 


43 


42 
41 


■84 394 


1500 


25 


4.685 57 


58 7 


-86 166 


5.314 43 


7-86 167 


1560 


26 


57 


58 


87 869 


43 


41 


• 87 871 


1620 


27 


57 


58 


89 508 


43 


41 


-89 510 


1680 


28 


57 


58 


91 088 


43 


41 


-91089 


1740 


29 
30 


57 


58 


92 612 


43 


41 


-92 613 


1800 


4. 685 57 


58 7 


94 084 


5-314 43 


41 


7-94 086 


1860 


31 


57 


58 


95 508 


43 


41 


-95 510 


1920 


32 


57 


53 


96 887 


43 


41 


-96 889 


1980 


33 


57 


59 


98 223 


43 


41 


-98 225 


2040 


34 
35 


57 


59 


99 520 


43 


41 


-99 522 


2100 


4.685 56 


59 8. 


00 778 


5.314 43 


41 


800 781 


2160 


36 


56 • 


59 


02 002 


43 


41 


-02 004 


2220 


37 


56 


59 


03 192 


43 


41 


-03 194 


2280 


38 


56 


59 


04 350 


43 


40 


-04 352 


2340 


39 


56 


59 


05 478 


43 


40 


-05 481 


2400 


40 


4. 685 56 


59 8 


06 577 


5.314 43 


40 


8-06 580 


2460 


41 


56 


59 


07 650 


43 


40 


-07 653 


2520 


42 


56 


59 


08 696 


43 


40 


-08 699 


2580 


43 


56 


60 


09 718 


43 


40 


-09 721 


2640 


44 


56 


60 


TO 716 


43 


40 
40 


-10 720 


2700 


45 


4. 685 56 


60 8. 


11 692 


5.314 44 


8-11 696 


2760 


46 


56 


60 


12 647 


44 


40 


.12 651 


2820 


47 


56 


60 


13 581 


44 


40 


.13 585 


2880 


48 


56 


60 


14 495 


44 


39 


.14 499 


2940 


49 


56 


60 


15 390 


44 


39 


-15 395 


3000 


50 


4-685 56 


60 8- 


16 268 


5.314 44 


39 


8-16 272 


3080 


51 


56 


60 


17 128 


44 


39 


.17 133 


3120 


52 


56 


61 


17 971 


44 


39 


-17 976 


3180 


53 


56 


61 


18 798 


44 


39 


-18 803 


3240 


54 
55 


55 


61 


19 610' 


44 


39 


. 7 Q '"15 


3300 


4.685 55 


61 8. 


20 407 


5.314 44 


39 


8-20 412 


3360 


56 


55 


6:; 


21 189 


44 


38 


-21 195 


£420 


57 


55 


6. 


21 958 


44 


38 


-21 964 


3480 


58 


55 


6: 


22 713 


44 


38 


.22 719 


3540 


59 


55 


62 


23 455 


44 


38 


23 462 



543 



TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. 



Log sin = log (ji" 4- S. 




r 


log 4>' 


' = log 


sin <i 4- *S'. 


Log tan <i> = log 4>" + T. 




log 0' 


' = log tan ^ + T'. 


It 


/ 


S 


T I.< 


3g:. Sin. 


S' 


T' 


Log. Tan. 


3600 





4.685 55 


62 8 


-24 185 


5-314 44 


38 


8.24 192 


3660 


1 


55 


62 


.24 903 


45 


38 


.24 910 


3720 


2 


55 


62 


.25 609 


45 


38 


.25 616 


3780 


3 


55 


62 


.26 304 


45 


37 


-26 311 


3840 


4 
5 


55 


62 


-26 988 


45 


37 


26 995 


3900 


4.685 55 


62 8 


-27 66l 


5-314 45 


37 


8-27 669 


3960 


6 


55 


63 


.28 324 


45 


37 


.28 332 


4020 


7 


54 


63 


.28 977 


45 


37 


-28 985 


4080 


8 


54 


63 


.29 620 


45 


37 


-29 629 


4140 


9 


54 


63 


30 254 


45 


36 


• 30 263 


4200 


10 


4.685 54 


63 8 


30 879 


5-31445 


36 


8-30 888 


4260 


11 


54 


63 


31 495 


& 


36 


-31 504 


4320 


12 


54 


64 


32 102 


45 


36 


-32 112 


4380 


13 


54 


64 


32 701 


46 


36 


-32 711 


4440 


14 


54 
4-685 54 


64 


33 292 


46 
5.314 46 


36 
35 


-33 302 


4500 


15 


64 8 


33 875 


833 885 


4560 


16 


54 


64 


34 450 


46 


35 


-34 461 


4620 


17 


54 


65 


35 018 


46 


35 


-35 029 


4680 


18 


54 


65 


35 578 


46 


35 


-35 589 


4740 


19 
30 


53 


65 


36 ]31 


46 


35 


■ 36 143 


4800 


4-685 53 


65 8 


36 677 


5.31446 


34 


836 689 


4880 


21 


53 


65 


37 217 


46 


34 


.37 229 


4920 


22 


53 


65 


37 750 


46 


34 


.37 762 


4930 


23 


53 


66 


38 276 


46 


34 


-38 289 


5040 


24 


53 


66 


38 796 


47 


34 


38 809 


5100 


25 


4685 53 


66 8 


39 310 


5.314 47 


33 


8-39 323 


5160 


26 


53 


66 


39 818 


47 


33 


-39 831 


5220 


27 


53 


67 


40 320 


47 


33 


.40 334 


5280 


28 


52 


67 


40 816 


47 


33 


-40 830 


5340 


29 


52 


67 


41 307 


47 


33 


■ 41 321 


5400 


30 


4.685 52 


67 8- 


41 792 


5-314 47 


32 


8-41 807 


5460 


31 


52 


67 


42 271 


47 


32 


.42 287 


5520 


32 


52 


68 


42 746 


47 


32 


.42 762 


5580 


33 


52 


68 


43 215 


48 


32 


.43 231 


5640 


34 


52 


68 


43 680 


48 


3l 


• 43 696 


5700 


35 


4-685 52 


68 8- 


44 139 


5-31448 


31 


8-44 156 


5760 


36 


52 


69 


44 594 


48 


31 


.44 611 


5820 


37 


51 


69 


45 044 


48 


31 


.45 06l 


5880 


38 


51 


69 


45 489 


48 


30 


.45 507 


5940 


39 


51 


69 


45 930 


48 


30 


• 45 948 


6000 


40 


4.685 5l 


69 8- 


46 366 


5.314 48 


30 


8^46 385 


6060 


41 


51 


70 


46 798 


49 


30 


.46 817 


6120 


42 


51 


70 


47 226 


49 


30 


.47 245 


6180 


43 


51 


70 


47 650 


49 


29 


.47 669 


6240 


44 


51 


70 


48 069 


49 


29 


■ 48 089 


6300 


45 


4-685 50 


71 8- 


48 485 


5-314 49 


29 


8-48 505 


6360 


46 


50 


7l 


48 896 


49 


28 


.48 917 


6420 


47 


50 


7T 


49 304 


4S 


28 


.49 325 


6480 


48 


50 


72 


49 708 


49 


28 


.49 729 


6540 


49 


50 


72 


50 108 


50 


28 
27 


-50 130 


6600 


50 


4.685 50 


72 8- 


50 504 


5.314 50 


8-50 526 


6660 


51 


50 


72 


50 897 


50 


27 


.50 920 


6720 


52 


50 


73 


51 286 


50 


27 


.51 310 


6780 


53 


49 


73 . 


51 672 


50 


27 


.51 696 


6840 


54 


49 


73 


52 055 


50 


26 


.52 079 


6900 


55 


4.685 49 


73 8- 


52 434 


5.314 50 


26 


8-52 458 


6960 


56 


49 


74 


52 810 


51 


26 


.52 835 


7020 


57 


49 


73 


53 183 


51 


25 


.53 208 


7080 


58 


49 


72 


53 552 


51 


25 


.53 578 


7140 


69 


49 


75 


53 918 


51 


25 


.53 944 



544 



TABLE VT— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES 



Log sin <j) = log (ji'' + S. 




2° 


log0' 


' = log sin 4> + <S'. 


Log tan 9 = log ^" + T. 


i 


log i>' 


' = log tan </) + T. 


II 


1 


1 

S 


T Log. Sin. 


S' 


T' Log. Tan. 


7200 





4-685 48 


75 8 


54 282 


5-314 51 


25 8 


54 308 


7260 


1 


48 


75 


54 642 


51 


24 


54 669 


7320 


2 


48 


75 


54 999 


5l 


24 


55 027 


7380 


3 


48 


76 


55 354 


52 


24 


55 381 


7440 


4 


48 


76 


55 705 


52 
5-314 52 


23 


55 733 


7500 


5 


4-685 48 


76 8 


56 054 


23 8 


56 083 


7560 


6 


48 


77 


56 400 


52 


23 


56 429 


7620 


7 


47 


77 


56 743 


52 


22 


56 772 


7680 


8 


47 


77 


57 083 


52 


22 


57 113 


7740 


9 


47 


78 


57 421 


52 
5-314 53 


22 


57 452 


7800 


10 


4-685 47 


78 8 


57 756 


22 8 


57 787 


7860 


11 


47 


78 


58 089 


53 


21 


58 121 


7920 


12 


47 


79 


58 419 


53 


21 


58 451 


7980 


13 


46 


79 


58 747 


53 


21 


58 779 


8040 


14 
15 


46 


79 


59 072 


53 
5.314 53 


20 


59 105 


8100 


4-685 46 


80 8 


59 395 


20 8 


59 428 


8160 


16 


46 


80 


59 715 


54 


20 


59 749 


3220 


17 


46 


80 


60 033 


54 


19 


60 067 


8280 


18 


46 


81 


60 349 


54 


19 


60 384 


8340 


19 


45 


81 


60 662 


54 
5.314 54 


19 


60 698 


8400 


30 


4.685 45 


81 8. 


60 973 


18 8- 


61 009 


8460 


21 


45 


82 


61 282 


54 


18 


61 319 


8520 


22 


45 


82 


61 589 


55 


18 


61 626 


8580 


23 


45 


82 


61 893 


55 


17 


61 931 


8640 


24 


45 


83 


62 196 


55 
5-314 55 


17 


62 234 


8Z00 


25 


4-685 44 


83 8- 


62 496 


16 8- 


62 535 


8760 


26 


44 


83 


62 795 


55 


16 


62 834 


8820 


27 


44 


84 


63 091 


55 


16 


63 133 


8880 


28 


44 


84 


63 385 


56 


15 


63 425 


89'4l3 


29 


44 


84 


63 677 


56 


15 


63 718 


9000 


30 


4-685 43 


85 8- 


63 968 


5.314 56 


15 8- 


64 009 


9080 


31 


43 


85 


64 256 


56 


14 


64 298 


9120 


32 


43 


86 


64 543 


56 


14 


64 585 


9180 


33 


43 


86 


64 827 


57 


14 


64 870 


9240 


34 


/i9 


. 86 


65 110 


57 
5.314 57 


13 


65 153 


9300 


35 


4.685 43 


87 8- 


65 391 


13 8. 


65 435 


9330 


36 


42 


87 


65 670 


57 


12 


65 715 


9420 


37 


42 


87 


65 947 


57 


12 


65 993 


9480 


38 


42 


88 


66 223 


58 


12 


66 269 


9540 


39 


42 


88 


66 497 


58 


IT 


66 543 


9600 


40 


4.685 42 


89 8- 


66 769 


5-314 58 


11 8- 


66 816 


9660 


41 


41 


89 


67 039 


58 


10 


67 087 


9720 


42 


41 


89 


67 308 


58 


10 


67 356 


9780 


43 


41 


90 


67 575 


59 


10 


67 624 


9840 


44 
45 


41 


90 


67 840 


59 


09 


67 890 


9900 


4-685 41 


91 8- 


68 104 


5-314 59 


C9 8 


68 154 


9960 


46 


40 


91 


68 360 


59 


08 


68 417 


10020 


47 


40 


9l 


68 627 


59 


08 


68 678 


10080 


48 


40 


92 


68 886 


60 


08 


68 938 


10140 


49 


40 


92 


69 144 


60 


07 


69 196 


10200 


50 


4-685 40 


93 8- 


69 400 


5-314 60 


07 8 


69 453 


10260 


51 


39 


93 


69 654 


60 


06 


69 708 


10320 


52 


39 


93 


69 907 


60 


06 


-69 961 


10380 


53 


39 


94 


70 159 


61 


06 


-70 214 


10440 


54 
55 


39 


94 


70 409 


61 


05 


-70 464 


10500 


4-685 38 


95 8 


70 657 


5-314 61 


05 8 


-70 714 


10560 


56 


38 


95 


70 905 


61 


04 


.70 962 


10620 


57 


38 


96 


71 150 


6l 


04 


.7120S 


10680 


58 


38 


96 


.71395 


62 


03 


.7145^ 


10740 


59 


38 


97 


.71638 


62 


03 1 


.71607 



545 



o^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



179' 



Log. Sin. 


— X 


6-46 372 


6 


76 475 


6 


94 084 


7 


06 578 


7 


16 269 


7 


24 187 


7 


30 882 


7 


38 681 


7 


41 797 


7 


46 372 


7 


50 512 


7 


54 290 


7 


57 767 


7 


60 985 


7 


63 981 


7 


66 784 


7 


69 417 


7 


71 899 


7 


74 248 


7 


76 475 


7 


78 594 


7 


80 614 


7 


82 545 


7 


84 393 


7 


86 166 


7 


87 869 


7 


89 508 


7 


91 088 


7 


92 612 


7 


94 084 


7 


95 508 


7 


96 887 


7 


98 223 


7 


99 520 


8 


00 778 


8 


02 002 


8 


03 192 


8 


04 350 


« 


nf\ ZL78 


8 


06 577 


8 


07 650 


8 


08 696 


8 


09 718 


R 


TO 716 


8 


11 692 


8 


12 647 


8 


13 581 


8 


14 495 


R 


15 390 


8 


16 268 


8 


17 128 


8 


17 971 


8 


18 798 


8 


19 610 


8 


20 407 


8 


21 189 ! 


8 


21 958 , 


8 


22 71^ 


8 


9.S /i'^'^ ! 


8 


24 185 


Log. Cos, j 



D 



30103 
17609 
12494 

9691 
7918 
6895 
5799 
5115 

4575 
4139 
3778 
3476 
3218 

2996 
2803 
2633 
2482 
2348 

2227 
2119 
2020 
1930 
1848 

1772 
1703 
1639 
1579 
1524 

1472 
1424 
1379 
1336 
1296 

1258 
1223 
1190 
1158 
1128 

1099 
1072 
1046 
1022 
998 

976 
954 
934 
914 
895 

877 
860 
8^3 
827 
811 

797 
782 
768 
755 
742 

730 



Log. Tan. 



D 



— 00 _ 

46 372 
76 475 
94 084 
06 578 



16 269 
24 188 
30 882 
36 681 
41 797 



46 372 
50 512 
54 291 
57 767 
60 985 



63 982 
66 785 
69 418 
71 900 
74 248 



76 476 
78 595 
80 615 
82 546 
84 394 



86 167 

87 871 
89 510 

91 089 

92 613 



94 086 

95 510 

96 889 

98 225 

99 522 



00 781 

02 004 

03 194 

04 352 

05 481 



06 580 

07 653 

08 699 

09 721 

10 720 



11 696 

12 651 

13 585 

14 499 

15 395 



16 272 

17 133 

17 976 

18 803 

19 815 



20 412 

21 195 

21 964 

22 719 

23 462 



24 192 



Com. D. 



Log.^Cot. 



30103 
17609 
12494 

9691 
7918 
6694 
5799 
5115 

4575 
4139 
3779 
3476 
3218 

2996 
2803 
2633 
2482 
2348 

2227 
2119 
2020 
1930 
1848 

1773 
1703 
1639 
1579 
1524 

1472 
1424 
1379 
1336 
1296 

1259 
1223 
1190 
1158 
1128 

1099 
1072 
1046 
1022 
999 

976 
954 
934 
914 
895 

877 
860 
843 
827 
812 
797 
783 
768 
755 
742 

730 



Com. D. 



Log. Cot. 


Log. Cos. 


+ 00 


00 000 


3 53 627 





00 000 


3 


•23 524 





00 000 


3 


• 05 915 





00 000 


p 


93 421 





00 000 


2 


83 730 





• 00 000 


2 


75 812 





.00 000 


2 


• 69 117 





.00 000 


2 


-63 318 





.00 000 


2 


-58 203 





-00 000 


2 


-53 627 





-00 000 


2 


.49 488 





-00 000 


2 


.45 709 


9 


• 99 999 


2 


-42 233 


9 


■ 99 999 


2 


-39 014 


9 


■ 99 999 


2 


-36 018 


9 


99 999 


2 


-33 215 


9 


-99 999 


2 


• 30 582 


9 


-99 999 


2 


-28 099 


9 


• 99 999 


9 


25 751 


9 


• 99 999 


2 


-23 524 


9 


-99 999 


2 


• 21 405 


9 


-99 999 


2 


-19 384 


9 


99 999 


2 


-17 454 


9 


-99 999 


2 


-15 605 


9 


• 99 999 


2 


-13 832 


9 


• 99 999 


2 


-12 129 


9 


• 99 999 


2 


-10 490 


9 


-99 998 


2 


-08 910 


9 


-99 998 


2 


-07 386 


9 


-99 998 


2 


-05 914 


9 


-99 998 


2 


-04 490 


9 


-99 998 


2 


-03 111 


9 


-99 998 


2 


-01 774 


9 


99 998 


2 


-00 478 


9 


99 998 




-99 219 


9 


• 99 997 




-97 995 


9 


• 99 997 




-96 805 


9 


• 99 997 




-95 647 


9 


• 99 997 




-94 519 


9 


• 99 997 




-93 419 


9 


• 99 997 




-92 347 


9 


• 99 997 




-91 300 


9 


99 997 




-90 278 


9 


-99 996 




-89 279 


9 


99 996 


I 


-88 303 


9 


-99 996 




.87 349 


9 


99 996 




-86 415 


9 


-99 996 




-85 500 


9 


99 996 




-84 605 


9 


99 995 




-83 727 


9 


99 995 




82 867 


9 


99 995 




82 023 


9 


99 995 




81 196 


9 


99 995 




80 384 


9 


99 994 




79 587 


9 


99 994 




78 804 


9 


99 994 




78 036 


9 


99 994 




77 280 


9 


99 994 




76 538 


9 


99 993 




75 808 


9 


99 993 


Lo 


g. Tan. 


Lc 


)g. Sin. 



89' 



546 



90' 



TABLE Vli— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 178° 



Log. Sin. 



24 185 

24 903 

25 609 

26 304 
26 988 



8.27 661 

8. 28 324 

8. 28 977 

8.29 620 
8-30 254 



8. 30 879 

8.31 495 

8.32 102 

8.32 701 

8.33 292 



33 875 

34 450 

35 018 

35 578 

36 131 



36 677 

37 217 

37 750 

38 276 
38 796 



8. 39 310 
8. 39 818 
8-40 320 
8-40 816 
8.41 307 



8-41 792 
8-42 271 

8. 42 746 

8.43 215 
8-43 680 



8.44 139 

8.44 594 

8.45 044 
8-45 489 
8 45 930 



8-46 366 

8. 46 798 

8.47 226 
8-47 650 
8-48 069 



8. 48 485 
8-48 896 

8.49 304 
8.49 708 
8-50 108 



8.50 504 

8.50 897 

8.51 286 
8.51 672 
8. 52 055 



52 434 

52 810 

53 183 
53 552 
53 918 



8.54 282 



Log. Cos. 



D 



71C 
706 
694 
684 

673 
663 
653 
643 
634 

625 
616 
607 
599 
591 

583 
575 
567 
560 
553 

546 
539 
533 
526 
520 

514 
508 
502 
496 
491 

485 

479 
474 
469 
464 

459 
454 
450 
445 
440 

436 
432 
428 
423 
419 

415 
411 
407 
404 
400 

396 
393 
389 
386 
382 

379 
375 
373 
369 
366 

363 



Log. Tan, 



24 192 

24 910 

25 616 

26 311 
26 995 



27 669 

28 332 

28 985 

29 629 

30 263 



30 888 

31 504 

32 112 

32 711 

33 302 



33 885 

34 461 

35 029 

35 589 

36 143 



36 689 

37 229 

37 762 

38 289 
38 809 



39 323 

39 83l 

40 334 

40 830 

41 321 



41807 
42 287 

42 762 

43 231 
43 696 



44 156 

44 611 

45 061 
45 507 
45 948 



46 385 

46 817 

47 245 

47 669 

48 089 



48 505 

48 917 

49 325 

49 729 

50 130 



50 526 

50 920 

51 310 

51 696 

52 079 



52 458 

52 835 

53 208 
53 578 
53 944 



8-54 308 



Log. Cot. 



Com. D. 



718 
706 
695 
684 

673 
663 
653 
643 
634 

625 
616 
607 
599 
591 

583 
575 
568 
560 
553 

546 
539 
533 
527 
520 

514 
508 
502 
496 
491 

485 
480 
475 
469 
464 
460 
455 
450 
445 
441 

437 
432 
428 
424 
419 

416 
412 
408 
404 
400 

396 
393 
390 
386 
383 
379 
376 
373 
370 
366 
364 



Com. D. 



Log. Cot. 



1.75 808 
1.75 090 
1.74 383 
1.73 688 
1-73 004 



1-72 331 
1.71 667 
1.71014 
1.70 371. 
1-69 736 



1.69 111 
1.68 495 
1.67 888 
1.67 288 
1.66 697 



1.66 114 
1.65 539 
1.64 971 
1.64 410 
1.63 857 



1.63 310 
1.62 771 
1.62 238 
1.61 711 
1.61 191 



1-60 676 
1.60 168 
1.59 666 
1.59 169 
1.58 678 



1.58 193 
1.57 713 
1.57 238 
1.56 768 
1-56 304 



1.55 844 
1.55 389 
1.54 938 
1 . 54 493 
1-54 052 



1.53 615 
1.53 183 
1.52 754 
1.52 330 
1-51 911 



Log. Cos. 



1.51 495 
1.51 083 « 
1.50 675 
1.50 270 
1-49 870 



1.49 473 
1.49 080 
1.48 690 
1.48 304 
1-47 921 



1.47 541 
1.47 165 
1.46 792 
1.46 422 
1-46 055 



1-45 691 



Log. Tan, 



99 993 
99 993 
99 993 
99 992 
99 992 



99 992 
99 992 
99 992 
99 991 
99 991 



99 991 
99 990 
99 990 
99 990 
99 990 



99 989 
99 989 
99 989 
99 989 
99 988 



99 988 
99 988 
99 987 
99 987 
99 987 



99 986 
99 986 
99 986 
99 986 
99 985 



99 985 
99 985 
99 984 
99 984 
99 984 



99 983 
99 983 
99 982 
99 982 
99 982 



99 981 
99 981 
99 981 
99 980 
99 980 



99 979 
99 979 
99 979 
99 978 
99 978 



99 978 
99 977 
99 977 
99 976 
99 976 



99 975 
99 975 
99 975 
99 974 
99 974 



99 973 



Log. Sin. 



91* 



547 



88' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



177' 



o 

1 
2 
3 
_4_ 

5 
6 
7 
8 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 

56 

57 

58 

59_ 

60 



93"* 



Log. Sin, 



8-54 282 
8.54 642 

8.54 999 

8. 55 354 
8-55 705 



8.56 054 
8-56 400 

8.56 743 

8.57 083 
8.57 421 



8.57 756 

8.58 039 
8.58 419 

8.58 747 

8.59 072 



8.59 395 

8.59 715 

8.60 033 
8.60 349 
8-60 662 



8.60 973 

8.61 282 
8.61 589 
8.61 893 
8-62 196 



8.62 496 

8.62 795 

8.63 091 
8.63 385 
8.63 677 



8.63 968 

8.64 256 
8.64 543 

8.64 827 
8-65 110 

8.65 391 
8.65 670 

8.65 947 

8.66 223 
8-66 497. 



8.66 769 

8.67 039 
8.67 308 
8.67 575 
8. 67 84-0 



8.68 104 
8.68 366 
8.68 627 
8.68 886 
8-69 144 



8-69 400 
8-69 654 
8-69 907 
8-70 159 
8-70 409 



8-70 657 
8-70 905 
8.71 150 
8-71 395 
8-71 638 



8-71 880 
Log, Cos, 



D 

360 
357 
354 
35l 

348 
346 
343 
340 
338 

335 
332 
330 
327 
325 

323 
320 
318 
316 
313 

311 
309 
306 
304 
302 

300 
298 
296 
294 
292 

290 
288 
286 
284 
282 

281 

279 
277 
275 
274 

272 
270 
268 
267 
265 

264 
262 
260 
259 
257 

256 
254 
253 
25l 
250 

248 
247 
245 
244 
243 

24l 



Log, Tan. 

8-54 308 

8.54 669 

8.55 027 
8.55 381 
8-55 733 



8-56 083 
8.56 429 

8.56 772 

8.57 118 
8-57 452 



57 787 

58 121 
58 451 

58 779 

59 105 



8-59 428 

8.59 749 

8.60 067 
8. 60 384 
8.60 698 



8-61 009 
8.61 319 
8-61 626 
8-61 931 
8-62 234 



8-62 535 

8.62 834 
8-63 131 

8.63 425 
8. 63 718 



8-64 009 
8-64 298 
8-64 585 
8-64 870 
8.65 153 



65 435 
65 715 

65 993 

66 269 
66 543 



8-66 816 
8.67 087 
8.67 356 
8.67 624 
8. 67 890 



68 154 
68 417 
68 678 

68 938 

69 196 



8. 69 453 
8- 69 708 

8. 69 961 

8.70 214 
8 . 70 464 



70 714 

70 962 

71 208 
71 453 
71 697 



8.71 939 



Log. Cot. 



Com. D. 



360 
358 
354 
352 

349 
346 
343 
341 
338 

335 
333 
330 
328 
325 

323 
320 
318 
316 
314 

311 
309 
307 
305 
303 

300 
299 
297 
294 
293 

291 
288 
287 
285 
283 

28l 
280 
278 
276 

274 

272 
271 
269 
267 
266 

264 
262 
261 
259 
258 

256 
255 
253 
252 
250 

249 
248 
246 
245 
243 
242 



Com. D. 



Log. Cot. 



1.45 691 
1.45 331 
1.44 973 
1.44 618 
1.44 263 



1.43 917 
1.43 571 
1.43 227 
1.42 886 
1.42 548 



Log. Cos. 



9. 99 973 
9.99 973 
9-99 972 
9.99 972 
9-99 971 



1.42 212 
1.41 879 
1.41 548 
1.41 220 
1.40 895 



1.40 571 
1.40 251 
1.39 932 
1.39 616 
1-39 302 



1.38 990 
1.38 681 
1.38 374 
1.38 068 
1.37 765 



1.37 465 
1.37 166 
1.36 869 
1.36 574 
1.36 281 



1.35 990 
1.35 702 
1.35 414 
1.35 129 
1.34 846 



99 971 
99 971 
99 970 
99 970 
99 969 



9-99 969 
9.99 968 
9.99 968 
9.99 967 
9.99 967 



9.99 966 
9.99 966 
9-99 965 
9.99 965 
9.99 964 



9.99 964 
9.99 963 
9.99 963 
9.99 962 
9. 99 962 



9.99 961 
9.99 961 
9.99 960 
9.99 959 
9.99 959 



1.34 565 
1.34 285 
1.34 007 
1.33 731 
1.33 456 



1.33 184 
1.32 913 
1.32 643 
1.32 376 
1.32 110 



1.31 845 
1.31 583 
1.31 32l 
1.31 062 
1.30 803 



30 547 
30 292 
30 038 
29 786 
29 535 



29 286 
29 038 
28 791 
28 546 
28 303 



9.99 958 
9.99 958 
9.99 957 
9.99 957 
P. 99 956 



9.99 956 
9.99 955 
9.99 954 
9.99 954 
9 .99 953 

9.99 953 
9. 99 952 
9. 99 952 
9. 99 951 
9. 99 950 



9.99 950 
9.99 949 
9. 99 948 
9. 99 948 
9. 99 947 



99 947 
99 946 
99 945 
99 945 
99 944 



9.99 943 
9. 99 943 
9.99 942 
9. 99 942 
9.99 941 



1.28 060 
Log. Tan, | 



9.99 940 



Log. Sin. 



548 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



176° 



Log. Sin. 



71 
72 
72 
72 
72 



880 
120 
359 
597 
833 



73 
73 
73 
73 
73 



069 
302 
535 
766 
997 

226 
453 
680 
905 
129 



75 
75 
75 
76 
76 



353 
574 
795 
015 
233 



451 
667 
883 
097 
310 



77 
77 
77 
78 
78 



522 
733 
943 
152 
360 

567 
773 
978 
183 
386 



79 
79 
79 
80 
80 



588 
789 
989 
189 
387 



80 
80 
80 
81 
81 



585 
782 
977 
172 
366 



81 
81 
81 
82 
82 



560 
752 
943 
134 
324 



513 
701 
888 
075 
260 

445 
629 
813 
995 
177 



84 



Log. 



358 
Cos. 



240 

239 
237 
236 
235 
233 
233 
231 
230 

229 
227 
226 
225 
224 

223 
221 
221 
219 
218 

217 
216 
215 
214 
213 
212 
211 
210 
209 
208 

207 
206 
205 
'204 
203 

202 
201 
200 
199 
198 

197 
197 
195 
195 
194 

193 
192 
191 
191 
189 
189 
188 
187 
186 
1185 

185 
184 
183 

182 
182 

181 
iT 



Log. Tan, 



939 
180 

420 



c.d. 



241 
240 
ocQi238 
111237 

366 nqO 

598|232 
831 ' oQi 
062 1 "^"^f 

525 228 
748^27 
^226 
||f,22_5 

422'!^ 



645 
867 
087 

306 

524 
741 
958 
172 
386 

599 
811 
022 
232 
441 



648 
855 
061 
266 
470 

673 
875 
076 
276 
476 

674 
871 
068 
264 
459 

653 
846 
038 
230 
420 

610 
799 
987 
175 
361 

547 
732 
916 
100 
282 



8. 84 464 



221 
220 
219 

218 
217 
216 
214 
214 

213 
212 
(210 
1210 
]209 

;207 
207 
206 
204 
204 

203 
202 
201 
200 
199 

198 
197 
197 
195 
195 

194 
193 
192 
191 
190 
190 
188 
188 
187 
186 

185 
185 
184 
183 
182 

182 



Log. Cot. 



Log. Cot. 



28 060 
27 819 
27 579 
27 341 
27 104 



26 868 
26 633 
26 400 
26 168 
25 937 



708 
479 
252 
026 
801 

57^ 
354 
133 
913 
693 



23 475 
23 258 
23 042 
22 827 
22 613 



22 400 
22 188 
21 978 
21 768 
21 559 



21 351 
21 144 
20 938 
20 734 
20 530 



20 327 
20 125 
19 923 
19 723 
19 524 



19 326 
19 128 
18 931 
18 736 
18 541 



18 347 
18 154 
17 961 
17 77C 
17 579 



17 389 
17 201 
17 012 
16 825 
16 638 



16 453 
16 268 
16 083 
15 900 
15 717 



Log. Cos. 



99 940 
99 940 
99 939 
99 938 
99 938 



99 937 
99 936 
99 935 
99 935 
99 934 



99 933 
99 933 
99 932 
99 931 
99 931 



99 930 
99 929 
99 928 
99 928 
99 927 



9 
9 
9 
9 
9 

9 
9 
9 
9 
9^^ 

9-99 919 
9. 99 918 
9.99 917 
9-99 916 
9. 99 916 



99 926 
99 B25 
99 925 
99 924 
j9 923 

99 92^ 
99 922 
99 921 
99 920 
99 919 



9.99 915 
9.99 914 
9.99 913 
9.99 912 
9-99 912 



99 911 
99 910 
9-99 909 
9. 99 908 
9-99 907 



9. 99 907 
9. 99 906 
9-99 905 
9.99 904 
9.99 903 



60 

59 
58 
57 
58 



50 

49 
48 

47 
46 



45 
44 
43 
42 

il 
40 

39 
38 

37 
36_ 
35 
34 
33 
32 
31 
30 
29 
28 
27 
2i 
25 
24 
23 
22 
21 



30 

19 
18 
17 
16 



15 535 



c.d.jLog, Tan 



99 902 10 

99 902 9 
99 901 8 
99 900 7 
99 899 6 

5 
4 
3 
2 

1 



9.99 898 
9.99 897 
9.99 896 
9. 99 896 
9. 99 895 



9-99 894 



Log. Sin. 
549 



P. P. 





330 


320 


310 


30( 


6 


33-0 


32-0 


31-0 


30. 


7 


38 


5 


37 


3 


36 


]. 


35. 


8 


44 





42 


6 


41 


3 


40. 


9 


49 


5 


48 





46 


5 


45. 


10 


55 





53 


3 


51 


6 


50. 


20 


110 





106 


6 


103 


3 


100. 


30 


165 





160 





155 





150. 


40 


220 





213 


3 


206 




200. 


50 


275 





266 


6 


258 


3 


250. 



290 

29.01 

33 

38 

43 

48 

96 
145 
193 
241 



380 

28.0 

32 

37 

42 

46 

93 
140 
186 
233 



370 

27.0 
31 



36 

40 

45 

90 

135 

180 

225 



360 

26.0 
30 



34 

39 

43 

86 

130 

173 

216 





350 


340 


330 


33( 


6 


25 


24.0 


23.0 


22 


7 


29 


1 


28.0 


26 


8 


25- 


8 


33 


3 


32.0 


30 


6 


29. 


9 


37 


5 


36.0 


34 


5 


33. 


10 


41 


6 


40.0 


38 


3 


36- 


20 


83 


3 


80.0 


76 


6 


73. 


30 


125 





120.0 


115 





110. 


40 


166 


6 


160.0 


153 


3 


146. 


50 


208 


3 


200.0 


191 


6 


183. 



310 

21 
24 
28 
31 
35 
70 
105 



6 

7 

8 

9 
10 
20 
30 
40 ; 140 
501175 



300 

20.0 

23.3 

26.6 

30.0 

33.3 

66-6 

100.0 

133-3 

166-6 



190 



19 
22 
25 
28 
31 
63 
95 
126 
158 



180 

18. 
21 



6 

7 

8 

9 

10 

20 

30 

40 

50 



8 

0.8 
0.9 



9 9 

0.910.9 

l-lll.O 

1.2il.2 

1.4 13 

1.6'l.5 

3.1J3.0 

4.7:4.5j4 

6.36.0l5 

7.9!7.5l6.6 



24 
27 
30 
60 
90 
61120 
31150 

6 5 



7 

8 
•1 
•Q 

1 

.3 

•513 
.6|4 
.815 



0.6f0 



0.7 
08 
0.9 





4 


4 


3 




3 


1 


0_ 


6 


0.4 


0.4 


0.3 


2 


0.1 


0.0 


7 


0.5 


0.4 





3 





2 


0.1 








8 


0.6 


0.5 





4 





2 


0.1 








9 


0.7 


0.6 





4 





3 


0.1 





1 


10 


0.7 


0.6 





5 





3 


0.1 





1 


20 


1-5 


1.3 


1 








6 


0-3 





T 


30 


2.2 


2.0 


1 


5 


1 





0-5 





2 


40 


3.0 


2.6 


2 





1 


3 


O.P 





3 


50 


3.7 


3-3 


2 


5 


1 


6 


08 





4 



P. p. 



S6' 



4° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



175" 



Log. Sin, 



358 
538 
718 
897 
075 

252 
429 
605 
780 
954 

128 
30l 
474 
645 
816 

987 

156 
325 
494 
661 

828 
995 
160 
326 
490 

654 
817 
980 

142 
303 



464 
624 
784 
943 
101 
25i 
417 
573 
729 
885 



040 
195 
349 
502 
655 
807 
959 
110 
261 
411 
561 
710 
858 
007 
154 

30l 
448 
594 
740 
885 



94 029 



|Log. Cos 



94^ 



180 
180 
178 
178 

177 
176 
176 
175 
174 

174 
173 
172 
171 
171 

170 
169 
168 
168 
167 
167 
166 
165 
165 
164 

163 
163 
162 
162 
161 

161 
16C 
159 
159 
158 

158 
157 
156 
156 
156 

155 
154 
154 
153 
153 

152 
151 
151 
150 
150 

150 
148 
148 
148 
147 

147 
146 
146 
146 
145 

144 

17 



Log. Tan, 



84 464 
84 645 

84 826 

85 005 
85 184 



85 363 
85 540 
85 717 

85 893 

86 068 



86 243 
86 417 
86 590 
86 763 
86 935 



87 106 
87 277 
87 447 
87 616 
87 785 



87 953 

88 120 
88 287 
88 453 
88 618 



88 783 

88 947 

89 111 
89 274 
89 436 



89 598 
89 759 

89 920 

90 080 
90 240 

90 398 
90 557 
90 714 

90 872 

91 028 



91 184 
91 340 
91 495 
91 649 
91 803 



91 957 

92 109 
92 262 
92 413 
92 565 



92 715 

92 866 

93 015 
93 164 
93 313 



93 461 
93 609 
93 756 

93 903 

94 049 

94 195 



Log. Cot. 



c.d. 

181 
180 
179 
179 

178 
177 
176 
176 
175 

175 
174 
173 
172 
172 

171 
170 
170 
169 
169 

168 

167 
167 
166 
165 

165 
164 
163 
163 
162 

162 
161 
161 
160 
159 

158 
158 
157 
157 
156 

156 
155 
155 
154 
154 

153 
152 
152 
151 
151 

150 
150 
149 
149 
149 

148 
148 
147 
146 
146 

145 
C.d. 



Log. Cot. 



1.15 535 
1.15 354 
1.15 174 
1.14 994 
1.14815 



14 637 
14 459 
14 283 
14 107 
13 931 



1.13 756 
1.13 582 
1.13 409 
1.13 237 
1.13 065 



12 893 
12 723 
12 553 
12 384 

12 215 



12 047 
11 880 
11 713 
11 547 
11 381 



11 216 
11 052 
10 889 
10 726 
10 563 



1.10 401 
1.10 24C 
1.10 079 
1.09 919 
1.09 760 



1.09 601 
1 . 09 443 
1.09 285 
1.09 128 
3 08 973 

1 
1 

1 
1 
1 



08 815 
08 660 
08 505 
08 350 
08 196 



1.08 043 
1.07 890 
1.07 738 
1-07 586 
1. 07 435 



07 284 
07 134 
06 984 
06 835 
06 686 



1.06 538 
1.06 390 
1.06 243 
1.06 097 
]^.05 950 
1 .05 805 
Log. Tan 



Log. Cos. 



99 894 
99 893 
99 892 
99 891 
99 890 



99 889 
99 888 
99 888 
99 887 
99 836 



99 885 
99 884 
99 883 
99 882 
99 881 



99 880 
99 879 
99 878 

99 877 
99 876 

99 875 
99 874 
99 874 
99 873 
99 872 



99 871 
99 870 
99 869 
99 868 
99 867 

99 866 
99 865 
99 864 
99 863 
99 862 



99 861 
99 860 
99 P59 
99 858 

99 857 



99 856 
99 855 
99 853 
99 852 
99 851 



99 850 
99 849 
99 848 
99 847 
99 846 



99 845 
99 844 
99 843 
99 842 

99 841 



99 840 
99 839 
99 837 
99 836 
_99 835 

9 • 99 834 
Log. Sin, 

550 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



40 

39 
38 
37 
36 



30 

29 
28 
27 

25 
24 
23 
22 
21 



20 

19 
18 
17 
16 



10 

9 
8 
7 



P. P. 





181 


180 


178 


17 


6 


6 


18.1 


18. 


17-8 


17.6 : 


7 


21 


1 


21 





20 


7 


20 


5 


8 


24 


1 


24 





23 


7 


23 


4 


9 


27 


1 


27 





26 


7 


26 


4 


10 


30 


1 


30 





29 




29 


3 1 


20 


60 


3 


60 





59 


3 


58 


6 1 


30 


90 


5 


90 





89 





88 





40 


120 


6 


120 





118 


6 


117 


3 


50 


150 


8 


150 





148 


3 


146 


6 11 





174 


173 


170 


168 


6 


17-4 


17-2 


170 


16-8 


7 


20 


3 


20 





19 


3 


19.6 


8 


23 


2 


22 


9 


22 


6 


22.4 


9 


26 


1 


25 


8 


25 


5 


25-2 


10 


29 





28 


6 


28 


3 


280 


20 


58 





57 


3 


56 


6 


56. 


30 


87 





86 





85 





84. 


40 


116 





114 


6 


113 


3 


112.0 


50 


145 





143 


3 


141 


6 


140-0 





158 


15 


6 


15 


4 


15 


6 


15. 8 


15-6 


15-4 


15 


7 


18 


4 


18 


2 


17 


9 


17 


8 


21 





20 


8 


20 


5 


20 


9 


23 


7 


23 


4 


23 


1 


22 


10 


26 


3 


26 





25 


6 


25 


20 


52 


6 


52 





51 


3 


50. 


30 


79 





78 





77 





76 


40 


105 


3 


104 





102 


6 


101 


50 


131 


6 


130 


c 


128 


3 


126- 



150 


149 


148 14 


150 


14-9 


14.81 14 


37 


5 


17 


4 


17 


21 17 


20 





19 


8 


19 


7| 19 


22 


r 


22 


3 


22 


2; 22 


25 


G 


24 


8 


24 


6; 24. 


50 





49 


6 


49 


3i 49 


75 





74 


5 


74 


0| 73- 


100 





99 


3 


98 


6l 98 


125 





124 


1 


123 


31122. 



146 

14-6 
17 
19 
21 
24 
48 
73 
97 
121 



145 

14-5 
16 



19 
21 
24 
48 
72 
96 
120 



110 

o.iio.i'o.o 

0.210.1 0.0 
0.2iO.I 0.0 
0.2 O.I 0.1 
0.2 O.IiO.l 
0.5 0.3 O.I 
0.70.5 0.2 
l.OiO.6 0.3 
1.2i0.8i0.4 



P. P. 





166 


164 


162 


160 ! 


6 


16. 6 


16.4 


16.2 


16.0 1 


7 


19 


3 


19 


1 


18 


9 


18.6 i 


8 


22 


1 


21 


8 


21 


6 


21 


3 


9 


24 


9 


24 


6 


24 


3 


24 





10 


27 


6 


27 


3 


27 





26 


6 


20 


55 


3 


54 


6 


54 





53 


3 


30 


83 





82 





81 





80 





40 


110 


6 


109 


3 


108 





106 


6 


50 


138 


3 


136 


6 


135 





133 





h 



TABLE VII.— LOGARITHMIC SINES, COSINES, 
AND COTANGENTS. 



TANGENTS, 



174' 



Log. Sin, 



8.94 029 
8-94 174 
8.94 317 
8.94 430 
8-94 603 



8-94 745 
8.94 887 
8-95 028 
8-95 169 
8-95 310 



8.95 450 
8.95 589 
8-95 728 
8-95 867 
8-96 003 



8.96 143 
8.96 280 
8-96 417 
8-96 553 
8.96 689 



8.96 825 
8.96 960 
8-97 094 
8-97 229 
897 363 



8-97 493 
8-97 629 
8-97 762 
897 894 
8-98 026 



8-98 157 

898 288 
8-98 419 
8-98 549 
8-98 679 

8-98 803; 
8-98 937 

899 066 
3-99 194 
8-99 322 



399 449 
8-99 577 
8-99 703 
899 830 
8-99 953 



144 
14cl 
143 
143 

142 
142 
141 
141 
140 

140 
139 
133 
133 
138 

133 
137 
137 
133 
133 

13d 
135 
13i 
134 
134 

133 
133 
132 
132 
132 

13l 
131 
133 
133 
130 

129 
123 
123 
128 
127 
127 



Log. Tan 



8-94 
8-94 
8-94 
8.94 
8.94 



195 
340 
485 
629 
773 



8.94 
8-95 
8-95 
8-95 
8-95 

8.95 
8-95 
8-95 
8-96 
8-98 



917 
059 
202 
344 
435 

626 
767 
907 
047 
1B6 



c.d. Log. Cot. Log. Cos 



8-96 
8.96 
8.96 



325 
464 
602 
8-96 739 
8-98 87^^ 



8-97 
8-97 
8-97 
8-97 
8-97 

8-97 
8-97 
8-97 
8-98 
8.98 



Ola 
149 
285 
421 
5-56 



690 
825 
958 
092 

225 



8-98 
8-98 
8-98 
8.98 
8-98 



8-99 
8-99 
8.99 
8-99 
8.99 



12^8.99 



9-00 081 
9-00 207 
9-00 332 
9-00 456 
9-00 580 



9-00 704 
9-00 828 
9-00 951 
9.01 073 
9.01 198 



9.01 318 
9.01 440 
9.01 561 
9.01 682 
9-01 803 



9-01 923 
[Log. Cos. 



123 
123 
123 

125 

125 
125 
124 
124 

124 
123 
123 
122 
122 

122 
122 
12l 
121 
120 

120 
T 



8-99 
8-99 
9-00 
9-00 



9-00 

9-00 

900 

9-00 

9-00^ 

9-00 

9-01 

9-01 

9-01 

9^1^ 

9-01 
9.01 
9-01 
9-01 
9:02 
9 02 



357 

490 
621 
753 
884 

015 
145 
275 
404 
53J 
662 
791 
919 
046 
174 

300 
427 
553 
379 
804 



933 
054 
179 
303 
427 

550 
673 
796 
918 
040 
162 



Log. Cot, 



145 
144 
144 
144 

143 
142 
142 
142 
141 

141 
141 
140 
140 
139 
139 
138 
133 
137 
137 

137 
138 
136 
135 
135 

134 
134 
133 
133 
133 

132 
132 
13l 
131 
131 

131 
130 
130 
129 
129 

129 
128 
123 
127 
127 

126 
126 
128 
125 
125 

125 
124 

124 
124 

124 

123 
123 
123 

122 
122 

121 

cTdT 



1.05 805 
1.05 659 
1-05 515 
1.05 370 
1.05 226 



1.05 083 
1.04 940 
1.04 798 
1.04 658 
1.04 514 



1.04 373 
1.04 232 
1.04 092 
1.03 952 
1.03 813 



9.99 834 
9-99 833 
9.99 832 
9.99 831 
9.99_830 

9.99 829 
9.99 827 
9.99 826 
9.99 825 
9.99124 

9.99 823 
9.99 822 
9.99 821 
9.99 819 
9.99 818 



1.03 674 
1.03 536 
1.03 398 
1.03 260 
1.03 12^3 



1.02 986 
1.02 850 
1.02 714 
1.02 57 
1.02 444 



1.02 309 
1.02 175 
1.02 041 
1.01 908 
1.01 775 



1.01 642 
1.01 510 
1.01 378 
1.01 247 
1.01 116 



9.99 799 
9.99 798 
9.99 797 
9.99 796 
9.99 794 



1.00 935 
1.00 855 
1.00 725 
1.00 595 
LOO 4.15 

1.00 337 
1.00 209 
1.00 08i 
0.99 953 
0-99 826 



0-99 699 
0-99 573 
0.99 446 
0-99 321 
099 195 



0-99 070 
0-98 945 
0-98 821 
0-98 697 
0-98 573 



9.99 817 
9.99 816 
9.99 815 
9.99 814 
9.99 813 



9.99 811 
9.99 810 
9-99 809 
9.99 808 
9.99 807 



9-99 805 
3-99 804 
3-99 803 
9.99 802 
9.99 801 



99 793 
99 792 

9.99 791 
99 789 

9-99 788 

9.99 787 
3.99 786 
3.99 784 
9.99 783 
9-99 782 



9-99 78i 
9.99 779 
9.99 778 
9.99 777 
9.99 776 



9-99 774 
9.99 773 
9.99 772 
9.99 770 
9.99 769 



0.98 450 
0.98 327 
0.98 204 
0-98 081 
0-97 959 



0-97 838 
Log. Tan, 



9.99 768 
9.99 766 
9.99 765 
9-99 764 
9-99 763 



60 

59 
58 
57 
-56 

55 
54 
53 
52 
-51 
50 
49 
48 
47 
43 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 

30 

29 
28 
27 
28 
25 
24 
23 
22 
21 

*?0 

19 
18 
17 



P. P. 



5"* 



9-99 761 
Log. Sin. 

551 



15 

14 

13 

12 

11 

10 

9 

8 

7 

_6 

5 

4 

3 

2 

_1 
O 





J4 


5 144 


143 142 14 


6 


14-51 14-4 


14-3 14-2 14- 


7 


16 


9! 16-8 


16. 7 16 


5 16- 


8 


19 


31 19-2 


19.0 18 


9 18- 


9 


21 


7 21.6 


21-4 21 


3 21- 


iO 


24 


i 24.0 


23-8 23 


6 23. 


20 


48 


3 48.C 


47.6 47 


3 47- 


30 


72 


51 72-0 


71.5 71 


70- 


40 


96 


61 96-0 


95-3 94 


6 94. 


50 


120. 81120. 


119-1118 


3 117- 



140 


139 


138 


137 136 


6 14-0 


13-9 


13-8 


13-7 13-6 


7 16-3 


16-2 


16 


1 


16-0 15.8 


8 18.6 


18-5 


18 


4 


18-2 18-1 


9 21-0 


20-8 


20 


7 


20-5 20-4 


10 23-3 


23-1 


23 





22-8 22-6 


20 46-6 


46-3 


46 





45-6 45-3 


30 70.0 


69-5 


69 





68-5 680 


40 93-3 


92-6 


92 





91-3 90-6 


50116.6 


115-8 


115 





114-1113-3 



135 134 133 



i 

9 
10 
20 
30 
40 
50 



13.5 


13.4 


13.3 


15-7 


15-6 


15-5 


18-0 


17-8 


17-7 


20-2 


20-1 


19-9 


22-5 


22-3 


22-1 


45-0 


44-6 


44-3 


67-5 


67-0 


66-5 


90-0 


89-3 


88-6 


112.5 


111-6 


110.8 





131 


130 


139 


6 


13-1 


13-0 


12-91 


7 


15-3 


15-1 


15 





8 


17-4 


17-3 


17 


2 


9 


19-6 


19-5 


19 


3 


10 


21-8 


21-6 


21 


5 


20 


43-6 


43-3 


43 





30 


65-5 


65-0 


6i 


5 


40 


87-3 


86-6 


86 





50 


109-1 


108-3 


107 


5 



133 

13-2 
15.4 
17-6 
19-8 
22-0 
44-0 
660 
88-0 
110.0 

138 

12-8 
14-9 
17-0 
19-2 
21.3 
42.6 
64.0 
85.3 
106.6 



137 

12-7 

14- 

16- 

19- 

21. 

42- 

63- 



84-6 



136 

12.5 
14-7 
16-8 
18. 9 
21-0 
42-0 
63-0 
84-0 



105-8 105-0 



135 

12-5 
14 



16 
18 
20 
41 
62 
83 
104 



134 

12-4 
14-4 
16-5 
18-5i 
20.6: 
41-3, 
62-0 
82-6 



133 

12-3 
14- 



103.3102 





133 


131 


6 


12-2 


12-1 


7 


14-2 


14-1 


8 


16.2 


16-: 


9 


18.3 


18-: 


10 


20.3 


20.1 


20 


40.6 


40.3 


30 


61.0 


60.5 


40 


81. 3 


80-6 


50 


101.6 


100-81 



130 

12-0 
14-0 
16-0 
18-0 
20-0 
40-0 
60.0 
80.0 



1 

O-T 
0-2 

0-2 

0-2 
0.2 
0.5 
0.7 

1-0 



1 

0-1 
0-1 
0-1 
0-1 
0-1 
0-3 
5 
0-6 



100-0 1-2 0-8 



o 

0-0 
0-0 
0-0 
0.1 
0.1 
O.I 
0-2 

o.S 

0-4 



P.P. 



84= 



TABLE VII.— LOGARITHMIC SINES, COSINES. TANGENTS, 

AND COTANGENTS. ±72^ 



Log. Sin, I d. 



o 

1 

2 
3 

_4_ 

5 
6 
7 
8 
_9^ 

10 

11 
12 
13 
1^ 

15 

16 

17 

18 

11 

20 

21 

22 

23 

21 

25 

26 

27 

23 

29_ 

30 

31 

32 

33 

31 

35 

36 

37 

38 

3± 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49_ 

50 

51 

52 

53 

54_ 

55 

56 

57 

58 

59 

60 



9.02 520 
9-02 638 
9-02 756 
9-02 874 
9.02 992 



9.03 109 
9.03 225 
9.03 342 
9-03 458 
9-03 574 



Log. Tan. c.d 



9.03 689 
9.03 805 

9.03 919 

9.04 034 
9.04 148 



9-04 262 
9.04 376 
9.04 489 
9.04 602 
9-04 715 



904 828 
9-04 940 
9-05 052 
9.05 163 
9.05 275 



9. 01 923.00 

9.02 043!tT5 
902 163!t|§ 
9.02 282i|t^ 
9.02 40ir^'^ 

119 
118 
118 
118 
;117 

1117 
jll6 
116 
'116 
|116 
115 
115 
114 
114 
114 

114 
113 
113 
113 
113 
112 
112 
112 
111 
111 

111 
110 

lie 
lie 

110 
109 
109 
109 
109 
108 

108 
108 
107 
107 
107 

107 
106 
106 
106 
106 

105 
105 
105 
104 
104 

104 
104 
103 
103 
103 

103 



9-02 162 
9.02 233 
9.02 404 
9.02 525 
9.02 645 



Log. Cot, 



9.02 765 

9. 02 885 

9.03 004 
9.03 123 
9.03 242 



9.03 361 
9-03 479 
9.03 597 
9.03 714 
9.03 83l 



9.05 386 
9-05 496 
9-05 607 
9.05 717 
9.05 827 



9-05 936 
9.06 046 
9.06 155 
906 264 
^•06 372 



9.06 480 
9.06 588 
9-06 696 
9-06 803 
906 910 



9.07 017 
9.07 124 
9-07 230 
9.07 336 
9-07 442 



9.07 548 
9.07 653 
9.07 758 
9.07 863 
9.07 967 



9.08 072 
9.08 176 
9-08 279 
908 383 
908 486 



9. 08 589 



Log, Cos. 



03 948 
9.04 065 
9.04 181 
9.04 297 
9.04 413 

9-04 528 
9-04 643 
9.04 758 
9.04 872 
9.04 987 



9. 05 101 
9-05 214 
9.05 327 
9-05 440 
9.05 553 



9.05 666 
9-05 778 

9.05 890 

9.06 001 
9.06 113 



9.06 224 
9.08 335 
9.06 445 
9.06 555 
9.08 665 



9.06 775 
9.06 884 

9.06 994 

9.07 102 
9.07 2ll 



9-07 319 
9.07 428 
9.07 535 
9.07 643 
9.07 750 



9.07 857 

9.07 964 

9.08 071 
9.08 177 
9.08 283 



9.08 389 
9.08 494 
9.08 600 
9.08 705 
9.08 810 



9. 08 914 



Log. Cot. 



121 
121 
120 
120 
120 
119 
119 
119 
119 

118 
118 
118 
117 
117 

117 
116 
116 
116 
115 

115 
115 
114 
114 
114 

114 
113 
113 
113 
113 

.112 
112 
112 
111 
111 

111 
111 
110 

lie 

110 

109 
1C9 
109 
109 
109 
108 
108 
107 
107 
107 

107 
107 
106 
106 
106 

105 
105 
105 
105 
105 

104 

C.d. 



0.97 838 
0.97 716 
0.97 595 
0.97 475 
0.97 354 

0.97 234 
0.97 115 
0.96 995 
0.96 876 
0.96 757 



Log, Cos. 



9.99 761 60 
9.99 760 59 
9.99 759 
9-99 757 
9.89 756 



y.99 754 
9.99 753 
9-99 752 
9-99 750 
9.99 749 



0.96 639 
0-96 521 
0-96 403 
096 285 
0.96 168 



0-96 051 
0-95 935 
0-95 818 
0-95 702 
0-95 587 



9. 99 748 
9.99 746 
9-99 745 
9.99 744 
9.99 742 



0.95 471 
0.95 356 
0.95 242 
0.95 127 
0-95 013 



0.94 899 
0.94 785 
0.94 672 
0.94 559 
0.94 446 



0.94 33*t 
0.94 222 
0.94 110 
0.93 998 
0.93 887 



0.93 776 
0.93 665 
0.93 554 
0.93 444 
0.93 3o4 



0.93 22o 
0.93 115 
0.93 006 
0.92 897 
0-92 788 



0-92 680 
0-92 572 
0.92 464 
0.92 357 
0-92 249 

0.92 142 
0.92 035 
0.91 929 
0.91 822 
0.91 716 



0.91 611 
0-91 505 
0.91 400 
0.91 295 
0.91 190 



9.99 741 
9. 99 739 
9. 99 738 
099 737 
Pfi9 735 



• 99 734 
9. 99 732 
9.99 73l 
9.99 730 
9.99 728 



9. 99 727 
9.99 725 
9-99 724 
9.99 723 
9.99 721 



9.99 720 
9.99 718 
9.99 717 
9.99 715 
9-99 714 



9. 99 712 
9.99 711 
9.99 710 
9.99 708 
9-99 707 



9.99 705 
9.99 704 
9.99 702 
9.99 701 
9.99 699 



9.99 698 
9.99 696 
9.99 695 
9-99 693 
9-99 692 



9.99 690 
9.99 689 
9.99 687 
9.99 686 
9-99 684 



091 085 



Log. Tan. 



9.99 683 
9-99 681 
9-99 679 
99 678 
9.99 676 



55 
54 
53 
52 
51 

50 

49 
48 
47 
46 



P.P. 



45 
44 
43 
42 
M. 
40 
39 
38 
37 
36 



35 
34 
33 
32 
31 

30 

29 
28 
27 
26 

25 
24 
23 

22 
21 

20 

19 
18 
17 
16 



15 
14 
13 
12 
_11 

10 

9 
8 
7 
6 



9.99 675 



96^ 



Log. Sin, 
552 



121 

12.1 
14.2 
16.2 
18.2 
20.2 
40.5 
60. 7i 
81.0 
101.2 



121 

12.1 
14.1 
161 
18.1 
20-1 
40.3 
60-5 
80.6 
100-8 



120 

12-0 
14 
16 
18 
20 
40 
60 
80 
100 



117_ 

6,11.7 
7il3.7 

8 15. 6 

9 17. 6 
10 19.6 
20 39.1 
30 58-7 
40 78.3 
50 97.9 



117 116 


11-7 11.61 


13.6 13.5 


15.6 15.4 


17.5 17.4' 


1I9.5 19.3 


39.0 38.6 


|58.5 58-0, 


78.0 77.3, 


97.5 96.6 



114 

11-4 
13.3 
15.2 
17.2 
19.1 
38.1 
57.2 
76-3 
95.4 



114 113 

111.411.3 



113.3 
il5.2 
I17.I 
I19.O 
!38.C 
57.0 
;76.0 
I95.O 



13.2 
15.0 
16.9 
18.8 
37.6 
56.5 
75.3 
94.1 



119 

11.9 
13.9 
15-8 
17.8 
19.8 
39.6 
59.5 
79.3 
991 

115 

11-5 
13.4 
15.3 
17.2 
19.1 
38-3 
57.5 
76-6 
95.8 



112 111 

11-2 11-1 

13-0112 ' 

14.9,14 

16.8il6 

18.6|l8 

37.3:37 

l56.0|55 

74.6174 

93.3192 



110 



6 

7 

8 

9 

10 

20 

30 

40 

50 



110 109 

11.0110.9 
12.8|12.7 
14.6 14.5 
16. 5 16. 3 
I8.3I18.I 
36.6;3b.3 
5b. 015^. 5 
73.1|7ii.6 
9i.6l9u.8 



108 

10.8 
12.6 
14.4 
16. 2 
18.0 
360 
5^0 
72.0 
9u.0 



6 

7 

8 

9 

10 

20 

30 

40 

50 



107 107 106 105 104 


10.7 


10.7 


10.6 


10.5 


10.4 


12.5 


12 


5 


12.3 


12.2 


12.1 


14.8 


14 


2 


14.1 


14.0 


13.8 


16.1 


16 





15.9 


15.7 


15. 6 


17.9 


17 


8 


17.6 


17.5 


17.3 


35.8 


35 


6 


35.3 


35.C 


34.6 


53.7 


53 


5 


53.0 


52-5 


52-0 


71.6 


71 


3 


70.6 


70-0 


69-3 


89.6 


89 


1 


883 


87.5 


86-6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



103 

10.3 
12.1 
138 
15.5 
17.2 
34.5 
51.7 
690 
86. 2 



103 

103 
12 



2 

0.2 
0.2 
0.2 
0.3 
0.3 
0.6 
1.0 
1.3 
1.6 



1 

0.1 
0.2 
0.2 
0.2 
0.2|0 



0.5 
0.7 
1.0 
1-2 



1 

0-1 
0.1 
O.I 
0.1 

1 

3 
5 
6 



0. 
0. 
0-_ 
0.8 



P.P. 



83' 



TABLE VII.— LOGARITHMIC SINES, COSINES, 
AND COTANGENTS. 



TANGENTS, 



172' 



8| 

51 1 

r 3 

) 6 
I 10 

11 

112 
!l3 
'ii 
115 

16 
■17 
1 18 

l± 

30 

21 

;2j 

^3 
2£ 

25 
26 

'27 

;28 
i2J__ 

30 

'31 

32 

33 
I 34 

1,35 

36 

37 

38 

39. 

10 

41 

42 
143 
[4_4 

1 45 
i46 
47 
48 
49^ 

50 

151 
52 
53 
51 
55 
56 
57 
58 
59^ 

60 



97' 



Log. Sin. I d. Log. Tan. c.d. Log. Cot 



08 589 
08 692 
08 794 
08 897 
08 999 



09 101 
09 202 
09 303 
9 09 404 
9 09 505 
9-09 606 
9. 09 706 
09 806 
09 908 
9-10 006 



10 lOo 
10 205 
10 303 
10 402 
10 501 



10 599 
9. 10 697 
9 10 795 
9. 10 892 
910 990 

9 
9 



11 087 
11 184 
11 281 
11377 
11 473! 



9. 11 570 
9-11 665 
9-11 761 
11 856 
11 952 



12 047 
9. 12 141 
912 236 
9-12 330 
9. 12 425 



12 518 
12 612 
12 706 
12 799 
12 892 



12 985 

13 078 
13 170 
13 263 
13 355 



13 447 
13 538 
9-13 630 
9-13 721 
913 813 



13 903 

13 994 

14 085 
14 175 
14 265 



914 355 
Log. Cos. 



102 
102 
102 
102 

102 
101 
101 
101 
101 

100 

100 

100 

100 

99 

99 

99 

98 

99 

98 

98 
98 
97 
97 
97 

97 
96 
Q7 
96 
96 

96 
95 



08 914 

09 018 
09 123 
09 226 
09 330 



09 433 
09 536 
9 09 639 
9-09 742 
9 . 09 844 



10 958 
9.11055 

11 155 
9-11 254 

11 353 



9-11 452 
9.11 550 
11 649 
9-11 747 
9.11 845 



95 
95 

95 
94 
94 
94 
94 

93 
94 
93 
93 
93 

93 
92 
92 
92 
92 

92 
9l 
92 
91 
91 

90 
91 
90 
90 
90 

90 



11943 
9 . 12 040 
12 137 
12 235 
12 331 



09 947 
9 10 048 
9-10 150 
9-10 252 
9-10 353 



9-10 454 
10 555 
9-10 655 
9-10 756 
910 85o 



104 
104 
103 
103 

103 

103 
103 
102 
102 

102 
lOl 
102 
101 
101 

101 
101 



0.91 085 
0.90 981 
0-90 877 
0-90 773 
0.90 670 



Log. Cos. 



0.90 566 
0.90 463 
0.90 360 
0.90 258 
0.90 155 



0.90 053 
0.89 951 
0-89 849 
0-89 748 
0-89 647 



0-89 546 
0-89 445 



99 675 
99 673 
9-99 672 
9.99 670 
9.99 669 



9.99 667 
9.99 665 
9.99 664 
9-99 662 
9-99 661 



}^S|o.89 344 
^^"'0.89 244 



12 428 
12 525 
12 621 
12 717 
12 813 



9-12 908 
9.13 004 
9.13 099 
9-13 194 
9-13 289 

13 384 
13 478 
13 572 
13 666 
13 760 



13 854 
13 947! 
9 14 041! 
9-14 134 
9-14 227 

9- 
9- 
9- 
9- 
9^ 

9 14 730 
Log. Cot. 



14 319 
14 412 
14 504 
14 596 
14 688 



100 

100 
99 
99 
99 
99 

98 
98 
98 
98 
98 
98 
97 
97 
97 
96 

97 
96 
96 
96 
96 

95 
95 
95 
95 
95 

94 
94 
94 
94 
94 

93 
93 
93 
93 
93 
92 
92 
92 
92 
92 

92 



0.88 548 
0.88 449 
0.88 351 
0.88 253 
0.88 155 



0.88 057 
0.87 959 
0.87 862 
0-87 765 
0.87 668 



9-99 659 
9-99 658 
9.99 656 
9-99 654 
9^99 653 
9 
9 



0-89 144 



89 044 
88 944 
0.88 845 
0.88 745 
0.88 646 



9.99 63!_ 
9-99 633 
9.99 632 
9.99 63g 
9.99 628 



0.87 571 
0-87 475 
0-87 379 
0-87 283 
0-87 187 



9.99 627 
9.99 625 
99 623 
99 622 
9-99 620 

9 

9 



0.87 091 
0.86 996 
0.86 900 
0-86 805 
0- 86 71Q 



86 616 
86 521 
86 427 
86 333 
86 239 



86 146 
86 052 
85 959 
85 866 
85 773 



085 680 
85 588 
0-85 495 
0-85 403 
0-85 311 



0-85 219 
Log, Tan. 



99 651 
99 650 
99 648 

99 646 

99 645 



9-99 643 
9.99 641 
9.99 640 
99 638 
9.99 637 



60 

59 
58 
57 

55 
54 
53 
52 
51_ 
50 
49 
48 
47 
46^ 

45 
44 
43 
42 
41 



99 618 
99 617 
-99 615 
-99 613 
-99 eii 



9-99 610 
9-99 608 

99 606 
9.99 605 

99 603 



9-99 601 
9-99 600 
9-99 598 
9-99 506 
9-99 594 



9-99 593 
9-99 591 
9-99 589 
9-99 587 
9^99 586 

9 

9 



99 584 
99 582 
9-99 580 
9. 99 579 
9 -99 577 
9-99 575 
Log. Sin. 

553 



40 

39 
38 
37 
36, 

35 
34 
33 
32 
11 
30 
29 
28 
27 
26_ 

25 
24 
23 
22 
21 



20 

19 
18 
17 
16 
15 
14 
13 
12 
11 



10 

9 
8 

7 
6 



P. P. 



104 103 103 101 

6 10-4 10-3 10 2 10-1 

7 12-1 12-0 11-9 11-8 

8 13-8 13-7 13. 6 13-4 

9 15-6 15-4,15-3 15-1 
10 17-3 17-1 17 16-8 
20 34-6 34-3:34-0 33-6 
30 52-0 51-5 51-0 50-5 
40 69-3 68-6 68-0 67-3 
50 86-6 85-3 85-0 84-1 





100 


100 


99 


9? 


6 


10-0 


10-0 


9-9! 9 


7 


11 


-7 


11 


6 


11 


5 


11 


8 


13 


4 


13 


3 


13 


2 


13- 


9 


15 


1 


15 





14 


8 


14. 


10 


16 


7 


16 


5 


16 


5 


16- 


20 


33 


5 


33 


3 


33 





32. 


30 


50 


-2 


50 





49 


5 


49- 


40 


67 





66 


6 66- 





65- 


50 


83 


7 


83 


3 


82- 


5 


81- 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



97 


91 




96 


9-7 


9-7 


9.6 


11 


4 


11 


3 


11-2 


13 





12 


9 


12.8 


14 


6 


14 


5 


14-4 


16 


2 


16 


1 


16-0 


32 


5 


32 


3 


32-0 


48 


7 


48 


5 


48-0 


65 





64 


6 


640 


81 


2 


80 


8 


80.0 



94 


94 


[ 


93 




93 


9.4 


9-4 


93 


9. 


11 





10 


9 


10 


8 


10- 


12 


6 


12 


5 


12 


4 


12- 


14 


2 


14 


1 


13 


9 


13- 


15 


7 


15 


6 


15 


5 


15- 


31 


5 


31 


3 


31 





30- 


47 


2 


47 





46 


5 


46- 


63 





62 


6 


62 





61. 


78 


7 


78 


3 


77 


5 


76. 



95 

9.5 
11.1 
12-6 
14-2 
15.8 
31.6 
47.5 
63-3 
79.1 





9T 


91 


90 


2 




6 


9.1 


9-1 


9.0 


0.2 


0- 


7 


10 


7 


10-6 


10 


5 


0.2 


0- 


8 


12 


2 


12-1 


12 





0-2 


0- 


9 


13 


7 


13-6 


13 


5 


0.3 


0. 


10 


15 


2 


15-1 


15 





0-3 


0. 


20 


30 


5 


30.3 


30 





0-6 


0- 


30 


45 


7 


45-5 


45 





1.0 


0. 


40 


61 





60.6 


60 





1-3 


1 


50 


76 


2 


75-8 


75 





1.6 


1. 



P. p. 



sn' 



8^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



171' 



O 

1 

2 

3 

_^ 

5 
6 
7 
8 

10 

11 
12 
13 
11 
15 
16 
17 
18 
19_ 

20 

21 
22 
23 
2^ 

25 

26 

27 

28 

29. 

30 

31 

32 

33 

34. 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44_ 

45 

48 

47 

48 

49_ 

50 

51 
52 
53 
5^ 

55 

56 

57 

58 

59_ 

60 



98' 



Log. Sin, 



914355 
9-14 445 
9-14 535 
9-14 624 
9.14 7ib 



14 802 
14 891 

14 980 

15 068 
15 157 



15 245 
15 333 
15 421 
15 508 
15 595 

15 683 
15 770 
15 857 

15 943 

16 030 



16 116 
16 202 
16 288 
16 374 
16 460 



16 545 
16 630 
16 716 
16 801 
16 88_5 

16 970 

17 054 
17 139 
17 223 
17 307 



17 391 
17 474 
17 558 
17 641 
17 724 



17 807 
17 890 

17 972 

18 055 
18 137 



18 219 
18 301 
18 383 
18 465 
18 546 



18 628 
18 709 
18 790 

18 871 
18952 

19 032 
19 113 
19 193 
19 273 
19 353 



9-19 433 



Log. Cos, 



90 
89 
89 
89 

89 
89 
88 
88 
88 

88 
83 
88 
87 
87 

87 
87 
87 
86 
86 

86 
86 
86 
86 
85 

85 
P5 
S'C 
85 
84 

84 
84 
84 
84 
84 
84 
83 
83 
83 
83 

83 
83 
82 
82 
82 

82 
82 
82 
81 
81 

8l 
81 
81 
80 

81 
80 
80 
80 
80 
80 

79 
T 



Log. Tan. c.d 



? 



14 780 
14 872 

14 983 

15 054 
15 14b 



15 236 
15 327 
15 417 
15 507 
15 598 



16 687 
15 777 
15 867 

15 956 

16 045 



16 134 
16 223 
16 312 
16 401 
16 489 



16 577 
16 665 
16 753 
16 841 
16 928 



17 015 
17 103 
17 190 
17 276 
17 363 



17 450 
17 536 
17 622 
17 708 
17 794 



17 880 

17 965 

18 051 
18 136 
18 221 



18 306 
18 390 
18 475 
18 559 
18 644 



18 728 
18 812 
18 896 

18 979 

19 063 



19 146 
19 229 
19 312 
19 395 
19 478 



19 560 
19 643 
19 725 
19 8C7 
19 889 



19 971 



Log. Cot 



91 
91 
91 
91 

91 
90 
90 
90 
90 

89 
90 
89 
89 
89 

89 
89 
89 
88 
88 

88 
88 
87 
88 
87 

87 
87 
87 
86 
87 

86 
86 
86 
86 
85 

86 
85 
85 
85 
85 

85 
84 
84 
84 
84 

84 
84 
84 
83 
83 

83 
83 
83 
83 
82 

82 
82 
82 
82 

82 
82 

cTd, 



Log. Cot. 



85 219 
85 128 
85 037 
84 945 
84 854 

84 763 
84 673 
84 582 
84 492 
84 402 



84 312 
84 222 
84 133 
84 043 
83 954 

83 865 
83 77b 
83 687 
83 599 
83 511 



83 422 
83 334 
83 247 
83 159 
83 071 



82 984 
82 897 
82 810 
82 723 
82 636 



82 550 
82 464 
82 37 
82 291 
82 206 



82 12G 
82 034 
81 949 
81 864 
81 779 



Log. Cos. 



81 60^ 
81 609 
81 525 
81 440 
81 356 



81 272 
81 188 
81 104 
81 020 
80 937 

80 854 
80 770 
80 687 
80 604 

80 522 

80 439 
80 357 
80 274 
80 192 
80 110 



0-80 028 
Log. Tan. 



99 575 
99 573 
99 571 
99 570 
99 568 



60 

59 
58 

57 
56 



99 566 
99 564 
99 563 
99 561 
99 559 



99 557 
99 555 
99 553 
99 552 
99 55C 



99 548 
99 546 
99 544 
99 542 
99 541 



99 539 
99 537 
99 535 
99 533 
99 53l 



99 529 
99 528 
99 526 
99 524 
99 522 



99 520 
99 518 
99 516 
99 514 
99 512 



99 511 
99 509 
99 507 
99 505 
99 503 



99 501 
99 499 
99 497 
99 495 
99 493 



99 491 
99 489 
99 487 
99 485 
99 484 

99 482 
99 480 
99 478 
99 476 
99 474 



99 472 
99 470 
99 468 
99 466 
99 464 

9-99 462 



Log, Sin 
554 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 

40 

39 
38 
37 
36 



30 

29 
28 
27 
26 

25 
24 
23 
22 

H 
20 

19 
18 
17 
16 



10 

9 
8 

7 
6 



P. P. 





91 


91 


90 


89 


6 


9.1 


9.1 


9.0 


89 


7 


10 


7 


10 


6 


10 


5 


10. 4 


8 


12 


2 


12 


1 


12 





11.8 


9 


13 


7 


13 


6 


13 


5 


13-3 


10 


15 


2 


15 


1 


15 





14.8 


20 


30 


5 


30 


3 


30 





29.6 


30 


45 


7 


45 


5 


45 





44.5 


40 


61 





60 


6 


60 





59.3 


50 


76 


2 


75 


8 


75. 





74.1 





88 


88 


87 


6 


8 8 


8 8 


8. 71 


7 


10 


3 


10 


2 


10 


1 


8 


11 


8 


11 


7 


11 


6 


9 


13 


3 


13 


2 


13 





10 


14 


7 


14 


6 


14 


5 


20 


29 


5 


29 


3 


29 





30 


44 


2 


44 





43 


5 


40 


59 





58 


6 


58 


.0 


50 


73 


7 


73 


3 


72 


.5 





85 


85 


84 


83 


6 


8.5 


85 


8.4 


8. 


7 


10 





9 


9 


9 


a 


9. 


8 


11 


4 


11 


3 


11 


2 


11. 


9 


12 


8 


12 


7 


12 


6 


12. 


10 


14 


2 


14 


1 


14 





13. 


20 


28 


5 


28 


3 


28 





27. 


30 


42 


7 


42 


5 


42 





41. 


40 


57 





56 




56 





55- 


50 


71 


2 


70 


8 


70 





69. 





83 


82 


81 


8< 


6 


8.2 


8.2 


8.1 


8. 


7 


9 


6 


9 


5 


9 


4 


9. 


8 


11 





10 


9 


10 


8 


10. 


9 


12 


4 


12 


3 


12 


1 


12 


10 


13 


7 


13 


6 


13 


5 


13 


20 


27 


5 


27 


3 


27 





26. 


30 


41 


2 


41 





40 


5 


40. 


40 


55 





54 


6 


54 





53. 


50 


68 


7 


68 


3 


67 


5 


66- 



86 
8.6 
ICO 
11.4 
12.9 
14.3 
28.6 
43. 
57.3 
71.6 





79 


2 


1 


6 


7.9 


0.2 


0. 


7 


9 


3 





2 


0. 


8 


10 


6 





2 


0. 


9 


11 


9 





3 


0. 


10 


13 


2 





3 


0. 


20 


26 


5 





6 


0- 


30 


39 


7 


1 





0. 


40 


53 





1 


3 


1. 


50 


66 


2 


1 


6 


1. 



p.p. 



81' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



170' 



Looj. Sin. 




d. 



223 
301 
379 
457 
535 

613 
690 
788 
845 
P^2 



999 
078 
152 
229 
305 



382 
458 
534 
&09 
685 



80 
79 
79 
79 

79 
79 
79 
78 
78 

78 
78 
78 
78 
! 78 

77 
77 
77 
77 
77 

77 
77 
76 
76 
76 

76 
76 
78 
75 
76 



761 
836 
911 
987 
062 

136 
2ll 
286 
360 
435 



509 
583 
657 
731 
805 



878 
952 
025 
098 
171 



244 
317 
390 
452 
535 

607 
679 
751 
823 
895 



23 967 



Log. Cos, 



75 
75 
75 
75 
75 
74 
75 
74 
74 
74 

74 
74 
74 
73 
74 

73 
73 
73 
73 
73 

73 
72 
73 
72 
72 

72 
72 
72 
72 
72 

71 
T 



Log. Tan. c.d- 



20 78i 
20 882 

20 942 

21 022 
21 102 



21 578 
21 657 
21 735 
21 814 
21 892 



19 971 

20 053; pT 

20 I34I cT 

20 216' 
20 297 



20 378 
20 459 

20 540 
20 820 
20 701 



81 

81 
81 
81 
80 

81 

80 
30 
80 
80 
80 

79 
79 
79 
79 
79 

79 
79 
78 
78 
78 

21 971: ll 

22 049' '^ 
22 127 
22 205 
22 283 



21 181 
21 261 
21 340 
21 423 
21 499 



22 360 
22 438 
22 515 
22 593 
22 670 



22 747 
22 824 
22 900 

22 977 

23 054 



23 130 
23 206 
23 282 
23 358 
23 434 



23 510 
23 588 
23 661 
23 737 
23 812 



23 887 

23 962 

24 037 
24 11^ 
24 186 

24 261 
24 335 
24 409 
24 484 
24 558 
24 632 



78 
78 
78 

77 
77 
77 
77 
77 

77 

77 
76 
76 
76 

76 
76 
76 
76 
76 

76 
75 
75 
75 
75 

75 
75 
75 
75 
74 

74 
74 
74 
74 
74 
74 



Loot. Cot, 



80 028 
79 947 
79 885 
79 784 
79 703 



79 622 
79 541 
79 460 
79 379 
79 298 



218 
138 
058 
978 
898 

818 
739 
659 
580 
501 



78 422 
78 343 
78 234 
78 188 
78 107 



78 029 
77 951 
77 873 
77 795 
77 717 



77 639 
77 582 
77 484 
77 407 
77 330 



77 253 
77 173 
77 099 
77 022 
76 946 



76 870 
76 793 
76 717 
76 641 
76 5^5 



Log. Cos, 



76 489 
76 414 
76 338 
76 263 
76 188 



76 11^ 
76 038 
75 963 
75 888 
75 813 



75 739 
75 664 
75 590 
75 51" 
75 A^2 



75 368 



Log. Cot. led.! Log. Tan 



99 462 
99 460 
99 458 
99 456 
99 454 
99 452 
99 450 
99 448 
99 446 
99 444 



99 442 
99 440 
99 437 
99 435 
99 433 



99 431 
99 429 
99 427 

99 425 
99 42R 

99 421 
99 419 
99 417 
99 415 
99 413 



99 411 
99 408 
99 406 
99 404 
99 402 



99 400 
99 398 
99 396 
99 394 
99 392 



99 389 
99 38_ 
99 385 
99 383 
99 381 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46. 
45 
44 
43 
42 

M, 
40 
39 
38 
37 
36 



30 

29 
28 
27 
26 



99 379 
99 377 
99 374 
99 372 
99 370 



99 368 
99 366 
99 364 
99 361 
99 359 

9^ 357 
99 355 
99 353 
99 350 
99 348 

99 346 
99 344 
99 342 
99 339 
9 9 337 
99 335 



Log. Sin 
555 



20 

19 
18 
17 
16 



10 

9 
8 

7 
_6^ 

5 
4 
3 
2 
1 
O 



P. P. 





81 




81 


80 


79 


6 


81 


8.1 


8.0 


7.9 


7 


9 


5 


9 


4 


9 


3 


9 


2 


8 


10 


8 


10 


8 


10 


6 


10 


5 


9 


12 


2 


12 


1 


12 


C 


11 


g 


10 


13 


6 


13 


5 


13 


3 


13 


1 


20 


27 


1 


27 





26 


6 


26 


3 


30 


40 


7 


40 


5 


40 





39 


5 


40 


54 3 


54 





53 


3 


52 


Q 


50 


67 


9 


67 


5 


66 


6 


65 


3 



78 
61 7 8 
7 9 
S'lO 



50!65 



78 
7 



1 13 

26 
39 
C2 
65 



77 
7.7 





76 


6 


7. 


7 


8 


8 


10. 


9 


11. 


10 


12. 


20 


25- 


30 


38- 


40 


51. 


50 


63. 



76 

7 
8 



10 
_ 11 
7il2 
5125 
2138 
0{50 
7!63 



75 

7.5 



74 

7-4 
6 
8 
1 
3 
8 

3 
6 



73 



7 
8 
9 

9'11 
IC 12 



73 

7 3 

8 5 



9 
10 
12 
24 
36 
0,48 
2 60 



71 

7 

8 

9 
10 



IGlll 
20J23 
30^35 
4047 
50159 



8 23 



72 

7 

8 

9 

10 

12 
24 
36 
48 
60 



2 

0.2 
0.2 
0-2 



1.1.6 



P P. 



80° 



10^ 



TABLE VII.— LOGARITHMIC SINES. COSINES, TANGENTS, 

AND COTANGENTS. 169® 



45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Log. Sin, 



967 
038 
110 
181 

252 



323 
394 
465 
536 
607 

677 
748 
818 
888 
958 



25 028 
25 098 
25 167 
25 237 
25 306 



25 376 
25 445 
25 514 
25 583 
25 652 



25 721 
25 790 
25 858 

25 927 
25J95 

26 063 
26 131 
26 199 
26 267 
26 335 



26 402 
26 470 
26 537 
26 605 
26 672 



26 739 
26 806 
26 873 

26 940 

27 007 



27 073 
27 140 
27 206 
27 272 
27 339 



27 405 
27 471 
27 536 
27 602 
27 668 



27 733 
27 799 
27 864 
27 929 

27 995 

28 060 



Log. Cos. d 



71 
71 
71 
71 

71 
71 
71 
71 
70 

70 
70 
70 
70 
70 

69 
70 
69 
70 
69 

69 
69 
69 
69 
69 

68 
69 
68 
68 
68 
68 
68 
68 
68 
67 

67 
68 
67 
67 
67 

67 
67 
67 
66 
67 

66 
66 
66 
66 
66 

66 
66 
65 
66 
65 

65 
65 
65 
65 
65 
65 



Log. Tan. 



G-24 
24 
24 
24 
24 



28 



632 
705 
779 
853 

926 

000 
073 
146 
219 
292 

365 
437 
510 
582 
654 

727 
799 
871 
943 
014 

086 
158 
229 
300 
371 

443 
514 
584 
655 
726 

796 
887 
937 
007 
078 

148 
218 
287 
357 
427 

496 
566 
635 
704 
773 

842 
9ll 
980 
049 
117 

186 

254 
322 
390 
459 

527 
594 
662 
730 
797 
865 



C. d. 

73 
74 
73 
73 

73 
73 
73 
73 
73 

73 
72 
72 
72 
72 

72. 
72 
72 
72 
7l 

72 
71 
71 
71 
71 

7l 
71 
70 
71 
70 

70 
70 
70 
70 
70 

70 
70 
69 
70 
69 

69 
69 
69 
69 
69 

69 
69 
68 
69 
68 

68 
68 
68 
68 
68 

68 
67 
68 
67 
67 

67 



Log. Cot. 



Log. Cot. c. d 



75 368 
75 294 
75 220 
75 147 
75 073 



75 000 
74 927 
74 854 
74 781 
74 708 



74 635 
74 562 
74 490 
74 417 
74 345 



74 273 
74 201 
74 129 
74 057 
73 985 



73 913 
73 842 
73 771 
73 699 
73 628 



73 557 
73 486 
73 415 
73 344 
73 274 



73 203 
73 133 
73 062 
72 992 
72 922 



72 852 
72 782 
72 712 
72 642 
72 573 



72 503 
72 434 
72 365 
72 295 
72 226 



72 157 
72 088 
72 020 
71 951 
71 882 



71 814 
71 746 
71 677 
71 609 
71 541 



71 473 
71 405 
71 337 
71 270 
71 202 



0.71 135 



Log. Tan. 



Log. Cos. 



99 335 
99 333 
99 330 
99 328 
99 326 



99 324 
99 321 
99 319 
99 317 
99 315 



99 312 
99 310 
99 308 
99 306 
99 303 



99 301 
99 299 
99 296 
99 294 
99 292 



99 29G 
99 287 
99 285 
99 283 
99 280 



99 278 
99 276 
99 273 
99 271 
99 269 



99 266 
99 264 
99 262 
99 259 
99 257 



99 255 
99 252 
99 250 
99 248 
99 245 



99 243 
99 240 
99 238 
99 236 
99 233 



99 231 
99 228 
99 226 
99 224 
99 221 



99 219 
99 216 
99 214 
99 212 
99 209 

99 207 
99 204 
99 202 
99 199 
99 197 



9. 99 194 



xoo= 



Log. Sin 
550 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41_ 

40 

39 
38 
37 
36 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21_ 

20 

19 
18 
17 
16 



10 

9 

8 

7 

_6 

' 5 
4 
3 
2 
1 



P. p. 



6 
7 
8 

9 
10 
20 
30 
40 
50 



74 

7-4 



73_ 

7-3 



73 

7-3 



72. 


72 


71 


71 


6 7.2 


72 


7.1 


71 


7 8 


4 


84 


8 


3 


8 


3 


8 9 


6 


96 


9 


5 


9 


4 


9 10 


9 


10.8 


10 


7 


10 


6 


10 12 


] 


12.0 


11 


9 


11 


3 


20 24 


1 


24-0 


23 


8 


23 


5 


30 36 


2 


36 


35 


7 


35 


5 


40 48 


3 


48.0 


47 


6 


47 


3 


50 60 


4 


60.0 


59 


6 


59 


I 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



70_ 

70 



70 

70 



69_ 

6-9 



68_ 
6-8 



68 
6-8 



8 
9 

10 
11 
23 
34 
46 
57 

67 



4 11 
22 



9 

2 
3 
6 
G 
3 
6156 



69 

6-9 



6 11 

1 23 

34 



7 
9 
C 

lilO 

2I1I 

5 22 

33 

44 



67 

6-7 



8 

9 

1 
3 
5 
6 
55 ? 





66 


66 


65 


65 


6 


6 6 


6-6 


6 5 


6-5 


7 


7 7 


7 


7 


7-6 


7 


6 


8 


8 


8 


8 


8 


8 


7 


8 


6 


9 


10 





9 


9 


9 


8 


9 


7 


10 


11 


1 


11 





10 


9 


10 


8 


20 


22 


1 


22 





21 


8 


21 


6 


30 


33 


2 


33 


C 


32 


7 


32 


5 


40 


44 


3 


44 





43 


6 


43 


f 


50 


55 


4 


55 





54 


6 


54 



6 
7 
8 

9 
10 
20 
30 
40 
50 



0-2 


0. 


0.3 


0. 


0.3 


0. 


0.4 


0. 


0.4 


0. 


0-8 


0. 


1.2 


1. 


1.6 


1. 


2.1 


1. 



2 

2 
2 
2 
3 
3 
6 

3 
6 



P. P. 



79' 



ir 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



168' 



' Log. Sin. d. Log. Tan. c. d 



9-28 060 
9. 28 125 
9.28 189 
9-28 254 
9-28 319 



9. 28 383 
9 . 28 448 
9-28 512 
9.28 576 
9 • 28 641 
9 
9 



28 705 
28 769 
28 832 
28 896 
28 960 



9-29 023 
9-29 087 
9.29 150 
9-29 213 
9.29 277 



29 340 
29 403 
9.29 466 
9-29 528 
9.29 591 
9 
9 



29 654 
29 716 
9.29 779 
9.29 841 
29 903 



9-29 965 
9.30 027 
9.30 089 
9.30 151 
9-30 213 



30 275 
30 336 
30 398 
30 459 
30 520 



9. 30 582 
9.30 643 
9.30 704 
9.30 765 
9. 30 826 



30 886 
30 947 
9. 31 008 
9-31 068 
9. 31 129 



9.31 189 
9-31 249 
9.31 309 
9.31 370 
9-31 429 



f -31 489 
9-31 549 
9. 31 609 
9. 31 669 
9.31 728 



9 31 783 
jLog. Cos. 



65 
64 
65 
64 

64 
64 
64 
64 
64 

64 
64 
63 
64 
63 

63 
63 
63 
63 
63 

63 
63 
63 
62 
63 

62 
62 
62 
62 
62 

62 
62 
62 
62 
6l 

62 
61 
61 
61 
61 

6l 
61 
61 
61 
61 

60 
61 
60 
60 
60 

60 
60 
60 
60 
59 

60 
60 

59 
60 
59 
59 



28 865 

28 932 

29 000 
29 067 
29 134 



9 
9 
9 
9 
9 

9. 29 201 
9-29 268 
9-29 335 
9-29 401 
9. 29 468 



9. 29 535 
9-29 601 
9. 29 667 
9-29 734 
^ 29 800 
9 
9 



29 866 
29 932 

9.29 998 

9.30 064 
9-30 129 



30 195 
30 260 
30 326 
30 391 
30 456 



9-30 522 
30 587 
9-30 052 
9-30 717 
9. 30 781 
9 ' 
9 



30 846 

30 911 
9-30 975 
9-31 040 

31 104 



9.31 168 
9.31 232 
31 297 
31 361 
31 424 



31 488 
31 552 
31 616 
31 679 
31 743 



9-31 806 
9-31 869 
9-31 933 
31 996 
9-32 059 



32 122 
32 ^85 
9-32 248 
9.32 310 
9-32 373 



9.32 436 
9-32 498 
9-32 580 
9 32 623 
9-32 685 



9 3? 747 
Log. Cot. 



67 
67 
67 
67 

67 
66 
67 
66 
67 

66 
66 
66 
66 
66 

66 
66 
66 
66 
65 

85 
65 
65 
65 
65 

65 
65 
65 
65 
64 

65 
64 
64 
64 
64 

64 
64 
64 
64 
63 

64 
64 
63 
63 
63 

63 
63 
63 
63 
63 

63 
63 
63 
62 
63 
62 
62 
62 
62 
62 

62 



Log. Cot. 



0-71 135 
0-71 067 
0.71 000 
0-70 933 
0.70 866 



0-70 798 
0-70 732 
070 665 
0.70 598 
0-70 531 



70 465 
70 398 
70 332 
70 266 
70 200 



Log. Cos, 



9-99 194 
9-99 192 
9.99 189 
9-99 187 
9-99 185 



9.99 182 
9.99 180 
9-99 177 
9-99 175 
9-99 172 



0.70 134 
0-70 068 
0.70 002 
0-69 936 
0-69 870 



0.69 805 
0-69 739 
0.69 674 
0.69 608 
0.69 543 



0.69 478 
0.69 413 
069 348 
0-69 283 
0-69 218 



9.99 170 
9-99 167 
ii.99 165 
9.99 162 
9.99 160 



9.99 157 
9.99 155 
9.99 152 
9.99 150 
9.99 147 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



9.99 145 
9.99 142 
9.99 139 
9.99 137 
9.99 134 



•99 132 
.99 129 
99 127 
99 124 
99 122 



45 
44 
43 
42 
_41 

40 

39 
38 
37 
36 



0.69 
0.69 
0.69 
0.68 
0-68 



0.68 
0.68 
0.68 
0.68 
068 



153 
089 
024 
960 
896 
831 
787 
703 
639 
575 



0-68 
0-68 
068 
0.68 
0.68 



0.68 
0-68 
0.68 
0.68 
0.67 



511 
447 
384 
320 
257 

193 
130 
067 
004 
941 



0.67 878 
0.67 815 
0.67 752 
0-67 689 
0.67 626 



0.67 56i 
0.67 501 
0.67 439 
0.67 377 
0-67 314 



067 252 
Log. Tan. 



99 119 30 
99 116 29 
99 114 28 



99 111 
99 109 



99 106 
99 104 
99 101 
99 098 
99 096 



99 093 
99 091 
99 088 
99 085 
99 083 



99 080 
99 077 
99 075 
99 072 
99 069 



99 067 
99 064 
99 062 
99 059 
99 056 



99 054 
99 051 
99 048 
99 046 
99 043 



9-99 040 
Log. Sin. 



25 
24 
23 
22 
21 

20 

19 
18 
17 
18 



10 

9 
8 
7 
6 



P. P. 



67_ 

67 



lOill 
20|22 



30|33 

40i450 

50i56-2 





66 


66 


65 


6/ 


6 


6.6 


6.6 


6.5 


6- 


7 


7 


7 


7 


7 


7 


6 


7. 


8 


8 


8 


8 


8 


8 


7 


8. 


8 


10 





9 


9 


9 


8 


9. 


10 


11 


1 


11 





10 


9 


10. 


20 


22 


1 


22 





21 


8 


21- 


30 


33 


2 


33 





32 


7 


32. 


40 


44 


3 


44 





43 


6 


43- 


50155 


4 


55 





54 


6 


54. 





64 


64 


63 


63 


6 


6.4 


6.4 


6.3 


6.3 


7 


7 


5 


7 


4 


7.4 


7 


3 


8 


8 


6 


8 


5 


84 


8 


4 


9 


9 


7 


9 


6 


9.5 


9 


4 


10 


10 


7 


10 


6 


10.6 


10 


5 


20 


21 


5 


21 


3 


21.1 


21 





30 


32 


2 


32 





31.7 


31 


5 


40 


43 





42 


6 


42-3 


42 





50 


53 


7 


53 


3 


52. gJ 


52 


5 



63 62 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 


2 


6 


.2 


6.1 


6 


7 


3 


7 


.2 


7.2 


7. 


8 


3 


8 


2 


82 


8. 


9 


4 


9 


3 


9.2 


9. 


10 


4 


10 


3 


10.2 


10. 


20 


8 


20 


6 


20.5 


20. 


131 


2 


31 





30.7 


30. 


41 


6 


41 


3 


41.0 


40. 


I52 


1 


51 


6 


51.2 


50. 



61 61 

1 
1 
1 
I 
I 
3 
5 
6 
8 



60 

6.0 



lO'lO 



60 

6-0 

70 

8.0 

9.0 

10-0 

20.0 

30.0 

40.0 

50-0 



59 

5.9 





3 


2 


6 


0.3 


0.21 


7 





3 





3 


8 





4 





3 


9 





4 





4 


10 





5 





4 


20 


1 








8 


30 


1 


5 


1 


2 


40 


2 





1 


6 


50 


9 


5 


9. 


1 



2 

0.2 
0.2 
0.2 
0.3 
0.3 
0.6 
l.Q 
1.3 
1.6 



P. P. 



557 



78' 



|ii 



13' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



16;^ 



Log. Sin. 



o 

1 

2 
3 

5 
6 
7 
8 

10 

11 

12 

13 

1^ 

15 

16 

17 

18 

19 



31 788 
31 847 
31 906 

31 966 

32 025 



32 084 
32 143 
32 202 
32 260 
32 319 



32 378 
32 436 
32 495 
32 553 
32 611 



20 

21 
22 
23 
24 

25 

26 

27 

28 

29l 

30 

31 

32 

33 

34, 

35 

36 

37 

38 

39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



32 670 
32 728 
32 786 
32 844 
32 902 



32 960 

33 017 
33 075 
33 133 
33 190 



33 248 
33 305 
33 362 
33 419 
33 476 



33 533 
33 590 
33 647 
33 704 
33 761 



33 817 
33 874 
33 930 

33 987 

34 043 



34 099 
34 156 
34 212 
34 268 
34 324 



34 379 
34 435 
34 491 
34 547 
•'34 602 



34 658 
34 713 
34 768 
34 824 
34 879 



34 934 

34 989 

35 044 
35 099 
35 154 



35 209 



Log. Cos. 



d. 



59 
59 
59 
59 

59 
59 
59 
58 
59 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
5? 
58 
57 
57 

57 
57 
57 
57 
57 

57 
57 
57 
57 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

55 
56 
55 
56 
55 

55 
55 
55 
55 
55 

55 
55 
55 
54 
55 

55 
T 



Log. Tan. 



32 747 
32 809 
32 871 
32 933 
32 995 



33 057 
33 118 
33 180 
33 242 
33 303 



33 364 
33 426 
33 487 
33 548 
33 609 



33 670 
33 731 
33 792 
33 852 
33 913 



33 974 

34 034 
34 095 
34 155 
34 215 



34 275 
34 336 
34 396 
34 456 
34 515 



34 575 
34 635 
34 695 
34 754 
34 814 



34 873 
34 933 

34 992 

35 051 
35 110 



35 169 
35 228 
35 287 
35 346 
35 405 



35 464 
35 522 
35 581 
35 640 
35 698 



35 756 
35 815 
35 873 
35 931 
35 989 



36 047 
36 105 
36 163 
36 221 
36 278 



36 336 



Log. Cot, 



C.d, 



62 
62 
62 
62 

61 
61 
62 
61 
6l 

61 
6l 
61 
61 
61 

60 
61 
61 
60 
61 

60 
60 
60 
60 
60 

60 
60 
60 
60 
59 

60 
60 
59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

58 
58 
59 
58 
58 

58 
58 
58 
58 
58 
58 
58 
57 
58 
57 

58 
c.d. 



Log. Cot. 



67 252 
67 190 
67 128 
67 066 
67 004 



66 943 
66 881 
66 819 
66 758 
66 696 



66 635 
66 574 
66 513 
66 452 
66 390 



66 330 
66 269 
66 208 
66 147 
66 086 



66 026 
65 965 
65 905 
65 845 
65 784 

65 724 
65 664 
65 604 
65 544 
65 484 



65 424 
65 364 
65 305 
65 245 
65 186 



65 126 
65 067 
65 008 
64 948 
64 889 



64 830 
64 771 
64 712 
64 653 
64 594 



64 536 
64 477 
64 418 
64 360 
64 302 



64 243 
64 185 
64 127 
64 068 
64 010 



63 952 
63 894 
63 837 
63 779 
63 721 



0-63 663 
Log. Tan. 



Log. Cos. 



99 040 
99 038 
99 035 
99 032 
99 029 



99 027 
99 024 
99 02l 
99 019 
99 016 



99 013 
99 010 
99 008 
99 005 
99 002 



98 999 
98 997 
98 994 
98 991 
98 988 



98 986 
98 983 
98 980 
98 977 
98 975 



98 972 
98 969 
98 966 
98 963 
98 961 



98 958 
98 955 
98 952 
98 949 
98 947 



98 944 
98 941 
98 938 
98 935 
98 933 



98 930 
98 927 
98 924 
98 921 
98 918 



98 915 
98 913 
98 910 
98 907 
98 904 



98 901 
98 898 
98 895 
98 892 
98 890 



98 887 
98 884 
98 881 
98 878 
98 875 



9-98 872 
Log. Sin. 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



40 

39 
38 
37 

35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



20 

19 
18 
17 
16 



10 

9 
8 
7 
6 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



62 

6-2 



61 

61 



61 

6.1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



60 

6.0 



60 

60 



59 

5 ■ 

6 

7 

8 

9 
19 
29 
39 
49 



59 

5-9 



6 

7 

8 

9 

10 

20 

30 

40 

50 



56 

5-6 



56 

56 
6 
7 
8 



55_ 

5-5 



55 

5-5 



6 

7 

8 

9 

10 

20 

30 

40 

50 



54 




3 


2 


5.4 


0-3 


0. 


6 


3 





3 


0- 


7 


2 





4 


0. 


8 


2 





4 


0. 


9 







5 





18 


■ 


1 





0. 


27 


2 


1 


5 


1. 


36 


3 


2 





1. 


45 


4 


2 


5 


2. 



P. r. 





58 


58 


57 


57 { 


6 


5.8 


5.8 


5-7 


57 \ 


7 


6 


8 


6 


7 


6 


7 


6 


6 f 


8 


7 


8 


7 


7 


7 


6 


7 


6 < 


9 


8 


8 


8 


7 


8 


6 


8 


5 


10 


9 


7 


9 


6 


9 


6 


9 


5 


20 


19 


5 


19 


3 


19 




19 





30 


29 


2 


29 





28 


7 


28 


5 


40 


39 





38 


6 


38 


3 


38 





50 


48 


7 


48 


3 


47 


9 


47 


5 



102*^ 



55S 



77 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



166' 



1 
2 
3 

5 

6 
7 

|9 

!io 
.1 

2 

'"3 

!:^ 

.6 
■.7 

1-8 
.9 



Log. Sin. 



35 209 
35 263 
35 318 
35 372 
35 427 



35 481 
35 536 
9.35 590 
9.35 644 
9.35 698 



9-35 752 
35 806 
35 860 
9-35 914 
9.35 968 



9. 36 021 
9-36 075 
9-36 128 
9. 36 182 
9.36 235 



9-36 289 
9-36 342 
9-36 395 
9-36 448 
9-36 501 



9-36 554 
9-36 607 
9-36 660 
9-36 713 
9-36 766 



36 818 
36 871 
9-36 923 
9-36 976 
9-37 028 



9-37 081 
9-37 133 
9-37 185 
9-37 237 
9-37 289 



9-37 341 
9-37 393 
9-37 445 
9-37 497 
9-37 548 



9.37 600 
9-37 652 
9-37 703 
9-37 755 
9-37 806 



37 857 
37 909 

37 960 

38 Oil 
38 062 



9. 38 113 
9-38 164 
9-38 215 
9-38 266 
9. 38 317 



9. 38 367 



54 
54 
54 
54 

54 
54 
54 
54 
54 
54 
54 
54 
53 
54 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
52 
53 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
5l 
52 
51 
52 
51 
51 
51 
51 

51 
5l 
51 
51 
51 

51 
51 
50 
51 
51 

50 



Log. Tan. c. d. Log. Cot. Log. Cos 



36 336 
36 394 
36 451 
36 509 
9. 36 566 



9-36 6231 
9-36 6811 
9-36 738! 
9-36 795 
9-36 852 



9-36 909 
9-36 966 
37 023 
37 080 
9-37 136 



37 193 
37 250 
37 306 
37 363 
37 419 



9-37 475 
9-37 532 
9-37 588 
9-37 644 
9.37 700 



9-37 756 
9-37 812 
9-37 868 
9-37 924 
9-37 979 



9-38 035 
9-38 091 
9-38 146 
9-38 202 
9. 38 257 



9. 38 313 
9-38 368 
9-38 423 
9-38 478 
38 533 



Log. Cos, d. 



38 589 
9-38 644 
9-38 698 
9-38 753 
9-38 808 



38 863 
38 918 

38 972 

39 027 
39 081 



39 136 
39 190 
9-39 244 
9-39 299 
9-39 353 



9.39 407 
9-39 461 
9-39 515 
9.39 569 
9.39 623 



939 677 
Log. Cot. 



5Z 
57 
57 
57 

5Z 
57 
157 
57 
57 

57 

5Z 
56 

5Z 
56 

5Z 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
55 
56 
56 
55 

56 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
54 
55 
55 

54 
55 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

53 



0-63 663 
63 606 
0-63 548 
0-63 491 
063 433 



063 376 
0-63 319 
0-63 262 
0-63 204 
0-63 147 



62 524 
62 468 
62 412 
62 356 
062 299 



0-63 090 
0-63 033 
0-62 977 
0-62 920 
0-62 863 



9-98 872 
9-98 869 
9-98 866 
9-98 863 
98 860 



062 806 
0-62 750 
0-62 693 
0-62 63Z 
0-62 580 



98 813 
98 810 
9-98 80Z 
9-98 804 
9-98 801 



62 243 
62 188 
62 132 
62 076 
62 020 



0-61 964 
0-61 909 
0-61 853 
0-61 798 
0-61 742 



0-61 687 
0-61 632 
0-61 576 
0-61 521 
0-61 466 



61 411 
61 356 
0-61 301 
061 246 
061 191 



61 137 
61 082 
61 027 
60 973 



9. 98 858 
9-98 855 
98 852 
98 849 
98 846 

98 843 
98 840 
98 837 
9-98 834 
9. 98 831 



60 

59 
58 
57 
56 



98 828 
98 825 
98 822 
98 819 
98 816 



98 798 
98 795 
98 792 
98 789 
98 786 



98 783 
98 780 
9-98 777 
9-98 774 
9-98 771 



9-98 768 

98 765 

9. 98 762 

9-98 759 

■98 755 



98 752 
98 749 
98 746 
98 743 
98 740 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 

40 

39 
38 
37 
36 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21, 

20 

19 
18 
17 
16 



060 918 



60 864 
60 809 
0-60 755 
060 701 
060 647 



060 592 
0-60 538 
0- 60 484 
0-60 430 
060 376 



98 737 
98 734 
98 731 
98 728 
98 725 



9-98 721 
9-98 718 
9-98 715 
9-98 712 
9-98 709 



9. 98 706 
9-98 703 
9-98 700 
98 696 
98 693 



0-60 323 
Log. Tan, 



9-98 690 
Log. Sin. 



P. p. 





57 


57 


56 


56 


6 


5.7 


5-7 


5-6 


5-6 


7 


6 


7 


6 


6 


6 


6 


6 


5 


8 


7 


6 


7 


6 


7 


5 


7 


4 


9 


8 


6 


8 


5 


8 


5 


8 


4 


10 


9 


6 


9 


5 


9 


4 


9 


3 


20 


19 


1 


19 





18 


3 


1.8 


^ 


30 


28 


7 


28 


5 


28 


2 


28 


Q 


40 


38 


3 


38 





37 


6 


37 


J^ 


50 


47 


9 


47 


5 


47 


1 


46 


6 





55 


55 


54 


54 


6 


5.5 


5-5 


5-4 


5.4 


7 


6-5 


6 


4 


6 


3 


6 


3 


8 


7.4 


7 


3 


7 


2 


7 


2 


9 


8 3 


8 


2 


8 


2 


8 


1 


10 


9-2 


9 


1 


9 


1 


9 





20 


18. 5 


18 


3 


18 


1 


18 





30 


27-7 


27 


5 


27 




27 





40 


37-0 


36 


6 


36 


3 


36 





50 


46-2 


45 


8 


45 


4 


45 








53 


53 


52 


53 


6 


5-3 


5.3 


5-2 


5-2 


7 


6 


2 


6 


2 


6 


1 


6 





8 


7 


1 


7 





7 





6 


9 


9 


8 





7 


9 


7 


9 


7 


8 


10 


8 


9 


8 


8 


8 


rr 
/ 


8 




20 


17 


8 


17 


6 


17 


5 


17 


3 


30 


26 


7 


26 


5 


26 


2 


26 





40 


35 


6 


35 


3 


35 





34 


5 


50 


44 


6 


44 


1 


43 


7 


43 


3 





51 


51 


50 


6 


5-1 


51 


50 


7 


6 





5 


.9 


5 


9 


8 


6 


8 


6 


8 


6 


7 


9 


7 


7 


7 


6 


7 


6 


10 


8 


6 


8 


5 8 


4 


20 


17 


1 


17 





16 


8 


30 


25 


7 


25 


5 


25 


2 


40 


34 


3 


34 





33 


g 


50 


42 


9 


42 


5 


42 


1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



3 

0-3 



3 2 

0-310.2 
0-3 0.3 
0-40.3 
0-40.4 
05 0.4 



10 
1.5 
2-0 
2-5 



0.8 
1.2 
1.6 
2.1 



P.P. 



03' 



559 



76' 



14^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 165^ 



Log. Sin. 



38 367 
38 418 
38 468 
38 519 
38 569 



38 620 
38 670 
38 720 
38 771 
38 821 



38 871 
38 921 

38 971 

39 021 
39 071 



39 120 
39 170 
39 220 
39 269 
39 319 



39 368 
39 418 
39 467 
39 516 
39 566 



39 615 
39 664 
39 713 
39 762 
39 811 



39 860 
39 909 

39 957 

40 006 
40 055 



40 103 
40 152 
40 200 
40 249 
40 297 



40 345 
40 394 
40 442 
40 490 
40 538 



40 586 
40 634 
40 682 
40 730 
40 777 



40 825 
40 873 
40 920 

40 968 

41 015 



41 063 
41 110 
41 158 
41 205^ 
41 252i 



41 299; 



d. 



50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

49 
50 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
48 
48 
49 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
47 
48 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 



Log. Cos.j d. 



Log. Tan. 



9-39 
39 
9 
9 
9 



677 
731 
784 
838 
8^2 

945 
999 
052 
106 
159 

212 
265 
318 
372 
425 



478 
531 
583 
636 
689 



742 
794 
847 
899 
952 

004 
057 
109 
161 
213 



266 
318 
370 
422 
474 

525 
577 
629 
681 
732 

784 
836 
887 
938 
990 

041 
092 
144 
195 

246 



297 
348 
399 
450 
501 

552 
602 
653 
704 
754 

805 



Log. Cot. 



c, d. Log. Cot, 



54 
53 
54 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
52 
53 
52 

53 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

51 
52 
52 
51 
51 

5l 
52 
5l 
51 
51 

51 
51 
51 
51 
5l 

51 
51 
51 
51 
50 

51 
50 
51 
50 
50 
50 

c. d. 



60 323 
60 269 
60 215 
60 161 
60 108 



054 
001 
947 
894 
841 
787 
734 
68l 
628 
575 



59 522 
59 469 
59 416 
59 363 
59 311 



59 258 
59 205 
59 153 
59 IOC 
59 048 



58 995 
58 943 
58 891 
58 838 
58 786 



58 734 
58 682 
58 630 
58 578 
58 526 



58 474 
58 422 
58 370 
58 319 
58 267 



58 216 
58 164 
58 112 
58 061 
58 OIC 



57 958 
57 907 
57 856 
57 805 
57 753 



57 702 
57 651 
57 600 
57 549 
57499 
57 448 
57 397 
57 346 
57 296 
57 245 



0-57 195 



Log, Tan, 



Log. Cos. 



S8 690 
98 687 
98 684 
98 681 
98 678 



98 674 
98 671 
98 668 
98 665 
98 662 



98 658 
98 655 
98 652 
98 649 
98 646 



98 642 
98 639 
98 636 
98 633 
98 630 



98 626 
98 623 
98 620 
98 617 
98 613 



98 610 
98 607 
98 604 
98 600 
98 597 



98 594 
98 591 
98 587 
98 584 
98 581 



98 578 
98 574 
98 571 
98 568 
98 564 



98 561 
98 558 
98 554 
98 551 
98 548 

98 544 
98 541 
98 538 
98 534 
98 531 



98 528 
98 524 
98 521 
98 518 
98 514 



98 511 
98 508 
98 504 
98 501 
98 498 



9. 98 494 



104^ 



Log. Sin, 
560 



60 

59 
58 
57 
_56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 

30 

29 
28 
27 
26 

25 
24 
23 
22 
21 

20 
19 
18 
17 
16 

15 
14 
13 
12 
11 

10 



P. P. 





54 


53 


53 , 


6 


5-4 


5.3 


5.3 - 


7 


6 


3 


6 


2 


6 


2 


8 


7 


2 


7 


2 


7 


Q 


9 


8 


1 


8 





7 


9 


10 


9 


C 


8 




8 


3 


20 


18 





17 


3 


17 


6 - 


30 


27 





26 


7 


26 


5 [ 


40 


36 





35 


Q 


35 


3 


50 


45 





44 


6 


44 


I 



52_ 

5-21 



53 

5-2 



40 35 
50143 



51 

5.1 



51 

5 

5 

6 

7 

8 
17 
25 
34 
42 



• 1 
.9 

• 8 

•5^ 
.0 

• 5 
.0 

5 





50 


50 


49 


49 : 


6 


5.0 


50 


4.9 


4.9. 


7 


5 


9 


5 


8 


5 


8 


5 


7 ' 


8 


6 


7 


6 


6 


6 


6 


6 


5 


9 


7 


6 


7 


5 


7 


4 


7 


3 ' 


10 


8 


4 


8 


3 


8 


2 


8 


I ' 


20 


16 


8 


16 


6 


16 


5 


16 


3 


80 


25 


2 


25 





24 


7 


24 


5 


40 


33 


g 


33 


3 


33 





32 


6 ■ 


50 


42 


1 


41 


6 


41 


2 


40 


8 





48 


48 


47 


47 


6 


4.8 


48 


4.7 


4. 


7 


5 


6 


5 


6 


5 


5 


5. 


8 


6 


4 


6 


4 


6 


3 


6. 


9 


7 


3 


7 


2 


7 


1 


7. 


10 


8 


1 


8 





7 


9 


7. 


20 


16 


1 


16 





15 


8 


15. 


30 


24 


2 


24 





23 


7 


23. 


40 


32 


3 


32 





31 


6 


31. 


50 


40 


4 


40 





39 


6 


39. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



0-3 








4 


0. 





4 


0. 





5 


0. 





6 


0. 


1 


1 


1. 


1 


7 


1. 


2 


3 


2. 


2 


9 


2. 



^! 15' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 164° 



7 

8 

J_ 

10 

11 
12 
13 
11 
15 
16 
17 
18 
19 



I 30 

' 21 
1 22 
> 23 
' 24 
25 
26 
f 27 
: 28 

; 29 

30 

I 31 
32 

: 33 

1 34 

I 35 

' 36 

37 

38 

I 39 



Log, Sin. 



9.41 299 
9.41 346 
9.41 394 
9-41441 
9-41 488 

9.41 534 
9. 41 581 
9.41 628 
9-41 675 
9-41 72] 



41 768 
41 815 
41 861 
41 908 
41 954 



d. 



9-42 000 
9-42 047 
9-42 093 
9-42 139 
9.42 185 



40 

41 
42 
43 
4£ 

45 
46 
47 
48 
49^ 

50 

51 
52 
53 
5£ 

55 
56 
57 
58 
59_ 
60 



42 232 
42 278 
42 324 
42 369 
42 415 



42 461 
42 507 
42 553 
42 598 
42 644 



42 690 
42 735 
42 781 
42 826 
42 87l 



42 917 

42 962 

43 007 
43 052 
4,"^ OOP 



43 143 
43 188 
43 233 
43 278 
43 322 



43 367 
43 412 
43 457 
43 501 
43 546 



43 591 
9-43 635 
9-43 680 
9-43 724 
9.43 768 



43 813 
43 857 
9. 43 90T 
43 945 
43 989 



9.44 034 



Log. Cos, 



47 
47 
47 
47 

46 
47 
47 
46 
46 

47 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
45 
46 

46 
46 
45 
45 
46 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
44 

45 
44 
45 
44 
44 

45 
44 
44 
44 
44 
44 
44 
44 
44 
44 

44 



Log. Tan. 



42 805 
42 856 
42 906 

42 956 

43 007 



43 057 
43 107 
43 157 
43 208 
43 258 



43 308 
43 358 
43 408 
43 458 
43 508 



43 557 
43 607 
43 657 
43 706 
43 756 



c.d. 



43 806 
43 855 
43 905 

43 954 

44 003 



44 053 
44 102 
44 151 
44 200 
44 249 



44 299 
44 348 
44 397 
44 446 
44 494 



44 543 
44 592 
44 641 
44 690 
44 738 



44 787 
44 835 
44 884 
44 932 
44 981 



45 029 
45 077 
45 126 
45 174 
45 222 



45 270 
45 318 
45 367 
45 415 
45 463 



45 510 
45 558 
45 606 
45 654 
45 702 



45 7^9 
Log. Cot. 



51 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

49 
50 
49 
49 
50 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
48 

49 
49 
48 
49 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

47 
48 
48 
47 
48 

47 
c.d. 



Log. Cot. 



57 195 
57 144 
57 094 
57 043 
56 993 



56 942 
56 892 
56 842 
56 792 
56 742 



0.56 692 
0.56 642 
0.56 592 
0.56 542 
0.56 492 



56 442 
56 392 
56 343 
56 293 
56 243 



0-56 194 
0.56 144 
0.56 095 
0.56 045 
0.55 996 



Log. Cos. d. 



55 947 
55 898 
55 848 
55 799 
55 750 



0.55 701 
0.55 652 
0.55 603 
55 554 
0.55 505 



55 456 
55 407 
55 359 
55 310 
55 261 



0.55 213 
0.55 164 
0.55 116 
0-55 067 
0-55 019 



54 970 
54 922 
54 874 
54 825 
54 777 



0-54 729 
0.54 681 
054 633 
0-54 585 
054 537 



0-54 489 
- 54 441 
0-54 393 
0-54 346 
0-54 298 



0-54 250 
Log. Tan. 



98 494 
98 491 
98 487 
98 484 
98 481 



98 477 
98 474 
98 470 
98 467 
98 464 



98 460 
98 457 
98 453 
98 450 
98 446 



98 443 
98 439 
98 436 
98 433 
98 429 



98 426 
98 422 
98 419 
98 415 
98 412 



98 408 
98 405 
98 401 
98 398 
98 394 



98 391 
98 387 
98 384 
98 380 
98 377 



98 373 
98 370 
98 366 
98 363 
98 359 



98 356 
98 352 
98 348 
98 345 
98 341 



98 338 
98 334 
98 331 
98 327 
98 324 



98 320 
98 316 
98 313 
98 309 
98 306 



98 302 
98 298 
98 295 
98 293 
98 288 



9-98 284 
Log. Sin. 



60 

59 
58 
57 
56 

55 
54 
53 
52 
11 
50 
49 
48 
47 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 

30 

29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 

17 

Ai 

15 
14 
13 
12 
11 



10 

9 

8 

7 

_6^ 

5 
4 
3 
2 
_], 




P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



50 



5 





5 


5 


9 


5. 


6 


7 


6. 


7 


6 


7. 


8 


4 


8. 


16 


8 


16. 


25 


2 


25. 


33 


6 


33. 


42 


1 


41. 



50 


8 
6 
5 
3 
6 

3 
6 





49 


49 


48 


48 


6 


4.9 


4.9 


4.8 


48 


7 


5 


8 


5 


7 


5.6 


5 


6 


8 


6 


6 


6 


5 


6.4 


6 


4 


9 


7 


4 


7 




7.3 


7 


2 


10 


8 


2 


8 


1 


8.1 


8 





20 


16 


5 


16 


3 


16.1 


16 





30 


24 


7 


24 


5 


24.2 


24 





40 


33 





32 




32.3 


32 





50 


41 


2 


40 


8 


40.4 


40 






47_ 

4-7 



30!23 
40131 
50l39 



47 

4.7 

5.5 

6.2 

7.0 

7-8 

15.6 

23.5 

31.3 

39.1 



46_ 

4.6 



46 

4.6 





45 


45 


44 


44 


6 


4-5 


4-5 


4-4 


4- 


7 


5 


3 


5 


2 


5 


2 


5. 


8 


6 





6 





5 


9 


5. 


9 


6 


8 


6 


7 


6 


7 


6- 


10 


7 


6 


7 


5 


7 


4 


7. 


20 


15 


1 


15 





14 


8 


14. 


30 


22 


7 


22 


5 


22 





22 


40 


30 


3 


30 





29 


6 


29. 


50 


37 


9 


37 


5 


37 


1 


36. 



3 

0.3 
0.4 
0.4 
0.5 
0.6 
1-1 
1-7 
2.3 
2.9 



3 

0.3 



P.P. 



106" 



561 



74' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
IG** AND COTANGENTS. 163" 



' Log. Sin. d. Log. Tan. c. d. Log. Cot. Log. Cos. d 



10 

11 

12 
13 
U 9 

15 
16 
17 
18 

ii 
20 

21 
22 
23 

2£ 

25 

26 

27 

28 

21 

30 

31 

32 

33 

34^ 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 

52 
53 

5^ 

55 
56 
57 
58 
59^ 

60 



44 034 
44 078 
44 122 
44 166 
44 209 



44 253 
44 297 
44 341 
44 381 
44 428 



44 472 
44 515 
44 559 
44 602 
44 646 



44 689 
44 732 
44 776 
44 819 
44 862 



44 905 
44 948 

44 991 

45 034 
45 077 



45 120 
45 163 
45 206 
45 249 
45 291 



45 334 
45 377 
45 419 
45 462 
45 504 



45 547 
45 589 
45 631 
45 674 
45 716 



45 758 
45 800 
45 842 
45 885 
45 927 



45 969 

46 Oil 
46 052 
46 094 
46 136 



46 178 
46 220 
46 261 
46303 
46 345 



46 38 
46 428 
46 469 
46 511 
46 55 2 

46*593 



Log. Cos. 



106' 



44 
44 
44 
43 

44 
44 
43 
43 
44 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 

42 
43 
43 
42 

42 
43 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
41 
42 
42 

4l 
42 
41 
41 
42 

41 
41 
41 
41 
41 

4l 



45 749 
45 797 
45 845 
45 892 
45 940 



45 987 

46 035 
46 082 
46 129 
46 177 



46 224 
46 271 
46 318 
46 366 
46 413 



46 460 
46 507 
46 554 
46 601 
46 647 



46 694 
46 741 
46 788 
46 834 
46_881 

46 928 

46 974 

47 021 
47 067 
47 114 



47 160 
47 207 
47 253 
47 299 
47 345 



47 392 
47 438 
47 484 
47 530 
47 576 



47 622 
47 668 
47 714 
47 760 
47 806 



47 851 
47 897 
47 943 

47 989 

48 034 



48 080 
48 125 
48 171 
48 216 
48 262 



48 307 
48 353 
48 398 
48 443 
48 488 



48 534 



Log. Cot. 



48 

47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
46 

47 
47 
46 
46 
47 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 
46 
46 
46 
46 
46 

46 
46 
45 
46 
46 

45 
46 
45 
46 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
C.d, 



54 250 
54 202 
54 155 
54 107 
54 060 



54 012 
53 965 
53 917 
53 870 
53 823 



53 776 
53 728 
53 681 
53 634 
53 587 



53 540 
53 493 
53 446 
53 399 
53 352 



53 305 
53 258 
53 212 
53 165 
53 118 



53 072 
53 025 
52 979 
52 932 
52 886 



52 839 
52 793 
52 747 
52 700 
52 654 



52 608 
52 562 
52 516 
52 469 
52 423 



52 377 
52 331 
52 286 
52 240 
52 194 



52 148 
52 102 
52 057 
52 Oil 
5] 965 



5] 920 
51 874 
51 829 
51 783 
51 738 



51 692 
51 647 
51 602 
51 556 
51'511 



0-51 466 
Log. Tan. 



98 284 
98 280 
98 277 
98 273 
98 269 



98 266 
98 262 
98 258 
98 255 
98 251 



98 247 
98 244 
98 240 
98 236 
98 233 



98 229 
98 225 
98 222 
98 218 
98 214 



98 211 
98 207 
98 203 
98 200 
98 196 



98 192 
98 188 
98 185 
98 181 
98 177 



98 173 
98 170 
98 166 
98 162 
98 158 



98 155 
98 151 
98 147 
98 143 
98 140 



98 136 
98 132 
98 128 
98 124 
98 121 



98 117 
98 113 
98 109 
98 105 
98 102 



98 098 
98 094 
98 090 
98 086 
98 082 



98 079 
98 075 
98 071 
98 067 
98 063 
98 059 



Log. Sin. 
562 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
-46 
45 
44 
43 
42 
41 

40 

39 
38 
37 

35 
34 
33 
32 

30 

29 
28 
27 

2e_ 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
"14 
13 
12 
11 

10 

9 
8 

7 
6 

5 
4 
3 

2 

T 

o 



p. p. 



48 47 47 



4 


8 


4 


7 


4. 


5 


6 


5 


5 


5. 


6 


4 


6 


3 


6. 


7 


2 


7 


1 


7 


8 





7 




7 


16 





15 


8 


15 


24 





23 


7 


23 


32 





31 


6 


31 


40 





39 


6 


39 





46 


46 


45 


45 


6 


4.6 


4.6 


4.5 


4-5 


7 


5 


4 


5 


.3 


5 


3 


5 


2 


8 


6 


.2 


6 


.1 


6 


.0 


6 





9 


7 


.0 


6 


• 9 


6 


8 


6 


7 


10 


7 


.7 


7 


• 6 


7 


■ 6 


7 


5 


20 


15 


• 5 


15 


• 3 


15 


. 1 


15 





30 


23 


• 2 


23 





22 


. 7 


22 


5 


40 


31 


• 


30 


. 6 


30 


• 3 


30 





50 


38. 7 


38. 3 


37-9 


37-5 


44 43 43 


6 


4 4| 


4.31 


4.3 


7 


5. 


1 


5. 


1 


5. 





8 


5. 


8 


5. 


8 


5- 


7 


9 


6. 


6 


6. 


5 


6. 


4 


10 


7. 


3 


7. 


2 


7. 


1 


20 


L4. 


6 


L4. 


5 


L4- 


3 


30 


22. 





21. 


7 


21. 


5 


40 


29. 


3 


29. 





28. 


6 


50, 


36.61 


36.21 


35. 8 


42 42 41 41 


6 


4.2 


4.2 


4.1 


41 


7 


4 


.9 


4 


9 


4 


.8 


4 


8 


8 


5 


.6 


5 


6 


5 


.5 


5 


4 


9 


6 


• 4 


6 


• 3 


6 


.2 


6 


1 


10 


7 


• 1 


7 





6 


9 


6 


8 


20 


14 


• 1 


14 





13 


.8 


13 


6 


30 


21 


.2 


21 


.0 


20 


•7 


20 


5 


40 


28 


.3 


28 


.0 


27 


■ 6 


27 


3 


50 


35 


.4 


35 


.0 


34 


6 


34 


1 



6 

7 
8 
9 
10 
20 
30 
40 
50 






4 








4 








5 








6 








6 





1 


3 


1. 


2 





1- 


2 


6 


2. 


3 


3 


2. 



P.P. 



M 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
17° AND COTANGENTS. 



162' 



Log. Sin. I d. 



46 593 
46 635 
46 676 
46 717 
46 758 



46 799 
46 840 
46 881 
46 922 
46 963 



47 0041 
47 045 j 
47 086 
47 127 
47 168 



47 208 
47 249 
47 290 
47 330 
47 371 



47 411 
47 452 
47 492 
47 532 
47 573 



47 6131 
47 653! 
47 694 
47 734 
47 774 



47 814 
47 854 
47 894 
47 934 
47 974 



48 014 
48 054 
48 093 
48 133 
48 173 



48 213 
48 252 
48 292 
48 331 
48 371 



48 410 
48 450 
48 489 
48 529 
48 568 



48 607 
48 646 
48 688 
48 725 
48 764 



48 803 
48 842 
48 881 
48 920 
48 959 



48 99fi 



I 41 

I 41 

41 

41 

41 
41 
41 
41 
41 

41 
41 
41 
40 
41 

40 
40 
41 
40 
40 

40 
40 

i 40 
40 

I 40 

' 40 
40 
40 
40 
40 

40 
40 
40 
40 
40 

40 
40 
39 
40 
39 
40 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

38 



Loe:. Cos, 



Log. Tan. 



48 534 
48 579 
48 624 
48 689 
48 714 



48 759 
48 804 
48 849 
48 894 
48 939 



48 984 

49 028 
49 073 
49 118 
49 162 



49 207 
49 252 
49 296 
49 341 
49 385 



49 430 
49 474 
49 518 
49 563 
49 607 



49 651 
49 695 
49 740 
49 784 
49 828 



49 872 
49 916 

49 960 

50 004 
50 048 



50 092 
50 136 
50 179 
50 223 
50 267 



50 311 
50 354 
50 398 
50 442 
50 485 



50 529 
50 572 
50 616 
50 659 
50 702 



50 746 
50 789 
50 832 
50 876 
50 919 



50 962 

51 005 
51 048 
51 091 
51 134 



51 17' 



Log. Cot. c. d. 



c.d, 

45 
45 
45 
45 

45 
45 
44 
45 
45 

45 
44 
45 
44 
44 

45 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 
44 
44 
43 
44 

44 
44 
43 
44 
43 
44 
43 
43 
44 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 



Log. Cot, 



51 486 
51 421 
51 376 
51 330 
51 285 



Log. Cos. d. 



51 240 
51 195 
51 151 
51 106 
51 061 



51 016 
50 971 
50 926 
50 882 
50 837 



50 792 
50 748 
50 703 
50 659 
50 614 



50 570 
50 525 
50 481 
50 437 
50 392 



50 348 
50 304 
50 260 
50 216 
50 172 



50 128 
50 083 
50 039 
49 996 
49 952 



49 908 
49 864 
49 820 
49 776 
49 733 



49 689 
49 645 
49 602 
49 558 
49 514 



49 471 
49 427 
49 384 
49 340 
49 297 



49 254 
49 210 
49 167 
49 124 
49 081 



49 038 
48 994 
48 951 
48 908 
48 865 



98 059 
98 056 
98 052 
98 048 
98 044 



98 040 
98 036 
98 032 
98 028 
98 024 



98 021 
98 017 
98 013 
98 009 
98 005 



98 001 
97 997 
97 993 
97 989 
97 985 



97 981 
97 977 
97 973 
97 969 
97 966 



97 962 
97 958 
97 954 
97 950 
97 946 



97 942 
97 938 
97 934 
97 930 
97 926 



97 922 
97 918 
97 914 
97 910 
97 906 



97 902 
97 898 
97 894 
97 890 
97 886 



97 881 
97 877 
97 873 
97 869 
97 885 



97 861 
97 857 
97 853 
97 849 
97 845 



97 841 
97 837 
97 833 
97 829 
97 824 



0-48 822 9-97 820 
Log. Tan.JLog. Sin. 

563 



60 

59 
58 
57 
56 



50 

49 
48 
47 
48 



40 

39 
38 
37 
36 



35 
34 
33 
32 
31 



30 

29 
28 

27 
28 



25 
24 
23 
22 
21 

20 

19' 
18 
17 
16 



10 

9 
8 
7 



P. P. 





4 


o 


45 


44 


44 


6 


4.5 


4-5 


4.4 


4-4 


7 


5 


3 


5 


2 


5 


2 


5 


1 


8 


6 





6 





5 


9 


5 


8 


9 


6 


8 


6 


7 


6 


7 


6 


6 


10 


7 


6 


7 


5 


7 


4 


7 


3 


20 


15 


1 


15 





14 




14 


6 


30 


22 


7 


22 


5 


22 


2 


22 





40 


30 


3 


30 





29 


6 


29 


3 


50 


37 


9 


37 


5 


37 


1 


36 


6 



43 



4.3 


4 


5 


1 


5 


5 


8 


5- 


6 


5 


6 


7 


2 


7 


14 




14. 


21 


7 


m 

Cl J. . 


29 





28. 


36 


2 


35. 



43 

3 


7 
4 
1 
3 
5 





41 


41 


40 


40 


6 


4.1 


4.1 


4.0 


4 


7 


4 


8 


4 


8 


4 


7 


4 


6 


8 


5 


5 


5 


4 


5 


4 


5 


3 


9 


6 


2 


8 


1 


6 


1 


6 





10 


6 


9 


6 


8 


6 


7 


6 


6 


20 


13 


8 


13 


6 


13 


5 13 


3 


30 


20 


7 


20 


5 


20 


2 20 





40 


27 


6 


27 


3 


27 


26 


6 


50 


34 


6 


34 


1 


33 


7 33 


3 





39 


39 


3 


6 


3.9 


39 


3 


7 


4 


5 


4 


5 


4 


8 


5 


2 


5 


2 


5 


9 


5 


9 


5 


8 


5 


10 


6 


6 


6 


5 


6. 


20 


13 


1 


13 





12. 


30 


19 


7 


19 


5 


19. 


40 


26 


3 


26 





25 


50 


32 


9 


32 


5 


32. 



50!3 



4 3 

040.3 



7.3 



r'. r. 



72** 



18"= 



TABLE VII —LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



161° 



Log. Sin. 



48 998 

49 037 
49 076 
49 114 
49 153 



49 192 
49 231 
49 269 
49 308 
49 346 



49 385 
49 423 
49 462 
49 500 
49 539 



49 577 
49 615 
49 653 
49 692 
49 730 



49 768 
49 806 
49 844 
49 882 
49 920 



49 958 

49 996 

50 034 
50 072 
50 110 



50 147 
50 185 
50 223 
50 260 
50 298 



50 336 
50 373 
50 411 
50 448 
50 486 

50 523 
50 561 
50 598 
50 635 
50 672 



50 710 
50 747 
50 784 
50 821 
50 858 



50 895 
50 932 

50 969 

51 006 
51 043 



51 080 
51 117 
51 154 
51 190 
51 227 



9.51 264 



Log, Cos, 



39 
39 
38 
39 

38 

39 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

33 
38 
38 
38 
38 

38 
38 
37 
38 
38 

37 
38 
37 
37 
38 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 
37 
37 
37 
37 
37 

37 
36 
37 
36 
37 
36 



Log. Tan 



51 177 
51 220 
51 263 
51 306 
51 349 



51 392 
51435 
51 477 
51 520 
51 563 



51 605 
51 648 
51 691 
51 733 
51 776 



51 818 
51 861 
51 903 
51 946 
51 988 



52 030 
52 073 
52 115 
52 157 
52 199 



52 241 
52 284 
52 326 
52 368 
52 410 



52 452 
52 494 
52 536 
52 578 
52 619 



52 661 
52 703 
52 745 
52 787 
52 828 



52 870 
52 912 
52 953 

52 995 

53 036 



53 078 

53 119 
53 161 
53 202 
53 244 



53 285 
53 326 
53 368 
53 409 
53 450 



53 491 
53 533 
53 574 
53 615 
53 656 



9-53 697 



Log. Cot. 



c. d, 

43 
43 
43 
43 
42 
43 
42 
43 
42 

42 
43 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
41 

42 
42 
4l 
42 
41 

4l 
42 
41 
41 
41 

4l 
4l 
41 
41 
41 

41 
41 
4l 
41 
41 

41 
4l 
41 
41 
41 

41 
C. d. 



Log. Cot. 



48 822 
48 779 
48 736 
48 693 
48 650 



608 
565 
522 
479 
437 



394 
351 
309 
266 
224 



48 181 
48 139 
48 096 
48 054 
48 012 



47 969 
47 927 
47 885 
47 842 
47 800 



47 758 
47 716 
47 674 
47 632 
47 590 



Log. Cos. 



47 548 
47 506 
47 464 
47 422 
47 380 



47 338 
47 296 
47 255 
47 213 
47 171 



47 130 
47 088 
47 046 
47 005 
46 963 



46 922 
46 880 
46 839 
46 797 
46 756 



46 714 
46 673 
46 632 
46 591 
46 549 



46 508 
46 467 
46 426 
46 385 
46 344 



0-46 303 



Log, Tan. 



97 820 
97 816 
97 812 
97 808 
97 804 



97 800 
97 796 
97 792 
97 787 
97 783 



97 779 
97 775 
97 771 
97 767 
97 763 



97 758 
97 754 
97 750 
97 746 
97 742 



97 737 
97 733 
97 729 
97 725 
97 721 



97 716 
97 712 
97 708 
97 704 
97 700 



97 695 
97 69l 
97 687 
97 683 
97 678 



97 674 
97 670 
97 666 
97 661 
97 657 



97 653 
97 649 
97 644 
97 640 
97 636 



97 632 
97 627 
97 623 
97 619 
97 614 



97 610 
97 606 
97 60l 
97 597 
97 593 

97 588 
97 584 
97 580 
97 575 
97 571 



9.97 567 



108^ 



Log, Sin. 
564 '. 



d. 



d. 



60 

59 
58 
57 
56 

55 
54 
53 
52 
-51 
50 
49 
48 
47 
46 

45 
44 
43 
42 
41 



40 

39 
38 
37 
M. 
35 
34 
33 
32 
_3I 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21_ 

20 

19 
18 

17 
16 

15 
14 
13 
12 
11 



10 

9 

8 

7 

_6, 

5 
4 
3 
2 
1 



P. P. 





43 


42 


42 


6 


4.3 


4.2 


4.2 


7 


5 





4 


9 


4 


9 


8 


5 


7 


5 


6 


5 


6 


9 


6 


4 


6 


4 


6 


3 


10 


7 


1 


7 


1 


7 





20 


14 


3 


14 


\ 


14 





30 


21 


5 


21 


2 


21 





40 


28 


6 


28 


3 


28 





50 


35 


8 


35 


4 


35 






6 

7 

8 

9 

•10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



41 



6 

7 

8 

9 

10 

20 

80 

40 

50 



4.1 


4. 


4 


8 


4. 


5 


5 


5 


6 


2 


6. 


6 


9 


6. 


13 


8 


13. 


20 


7 


20. 


27 


6 


27. 


34 


6 


34. 



41 

1 
8 
4 
I 
8 
6 
5 
3 
1 



39 

39 



38 

3.8 



38 

38 



37 

3.7 

4.4 

5.0 

5.6 

6.2 

12-5 

18.7 

25.0 

31.2 



37 

3.7 

4.3 

4-9 

5.5 

6.1 

12.3 

18-5 

24.6 

30.8 



36 

3-6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



4 

0.4 
0.5 
0.6 
0.7 
0.7 
1.5 
2-2 
3.0 
3.7 



P. P. 



ir 



J 



TABLE VII.— LOGARITHMIC SINES, COSINES, 
AND COTANGENTS. 



TANGENTS, 



160' 



Log. Sin. 



51 264 
51 301 
51 337 
51 374 
51 410 



51 447 
51 483 
51 520 
51 556 
51 593 



51 629 
51 665 
51 702 
51 738 
51 774 



51 810 
51 847 
51 883 

51 919 
51 955 



51 991 

52 027 
52 063 
52 099 
52 135 



52 170 
52 206 
52 242 
52 278 
52 314 



52 349 
52 385 
52 421 
52 456 
52 492 



52 527 
52 5631 
52 598 
52 634! 
52 869! 



52 704 
52 740 
52 775 
52 810 
52 846 



52 881 
52 916 
52 951 

52 986 

53 021 



53 056 
53 091 
53 126 



37 
36 
36 
36 

36 
36 
36 
36 
36 

36 
33 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
36 

35 
36 
36 
35 
36 

35 
35 
36 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 



53 16l' ^^ 



53 196 



53 231| 
53 266 
53 301 
53 335 
53 3701 



53 4051 



Log. Cos.j d. 



Log. Tan. 



53 697 
53 738 
53 779 
53 820 
53 861 



53 902 
53 943 

53 983 

54 024 
54 065 



54 106 
54 147 
54 187 
54 228 
54 269 



54 309 
54 350 
54 390 
54 431 
54 471 



54 512 
54 552 
54 593 
54 633 
54 673 



54 714 
54 754 
54 794 
54 834 
54 874 



54 9151 
54 955; 

54 995' 

55 035' 
55 075' 



55 115~i 
55 1551 
55 195- 
55 235! 
55 275; 



55 315, 



55 434! 
55 474' 



55 514 
55 553 
55 593 
55 633 
55 672 



55 712 
55 751 
55 79l 
55 831 
55 870 



55 909 
55 949 



c.d.iLog, Cot, 



41 
41 
41 
41 

41 
41 
40 
41 
41 

40 
41 
40 
40 
41 

40 
40 
40 

40 
40 

40 
40 
40 
40 
40 

40 
40 
40 
40 
40 

40 
40 
40 
40 
40 
40 
39 
40 
40 
40 

40 
40 



55 3551 05 
55 394' ^^ 



55 9881 05 

56 028! i^ 
56 067 1 "^^ 
56 1061 ^^ 



Log. Cot. c. d. 



46 303 
46 262 
46 221 
46 180 
46 139 



46 098 
46 057 
46 016 
45 975 
45 934 



45 894 
45 853 
45 812 
45 772 
45 731 



45 690 
45 65C 
45 609 
45 56S 
45 528 



9 
9 
9 
9 
9 
45 286 9 



45 488 
45 447 
45 407 
45 367 
45 326 



45 246 
45 205 
45 165 
45 12e 



Log. Cos, 



45 085 
45 045 
45 005 
44 965 
44 925 



44 884 
44 845 
44 805 
44 765 

44 72f: 



44 885 
44 845 
44 605 
44 565 
44 526 



44 488 
44 446 
44 406 
44 367 
44 327 



44 288 
44 248 
44 208 
44 169 
44 129 



44 09G 
44 051 
44 011 
43 972 
43 932 



43 893 



Log, Tan 



97 567 
97 562 
97 558 
97 554 
97 549 



97 545 
97 541 
97 536 
97 532 
97 527 



97 523 
97 519 
97 514 
97 510 
97 505 



97 501 
97 497 
97 492 
97 488 
97 483 



97 479 
97 475 
97 470 
97 466 
97 461 



97 457 
97 452 
97 448 
97 443 
97 430 



97 434 
97 430 
97 425 
97 421 
97 416 



97 412 
97 407 
97 403 
97 398 

Q7 394 



97 389 
97 385 
97 380 
97 376 
97 371 



97 367 
97 362 
97 358 
97 353 
97 349 



97 344 
97 340 
97 335 
97 330 
97 326 



97 321 
97 317 
97 312 
97 308 
97 303 



97 298 



Log. Sin. 
565 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 

45 
44 
43 
42 

11 
40 

39 
38 
37 
18 

35 
34 
33 
32 
31 



30 
29 
28 
27 

2e 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 



P. p. 



41 

4.1 
48 
5-4 
61 
6.8 
13-6 
20.5 
40127.3 
50134.1 



40_ 

401 



40 

40 



39 



3 


9 


3. 


4 


6 


4. 


5 


2 


5. 


5 


9 


5. 


6 


6 


6. 


13 


1 


13. 


19 


7 


19- 


26 


3 


26. 


32 


9 


32. 



39 

9 
5 
2 
8 
5 

5 

5 





37 


36 


3C 


6 


3.7 


3.6 


3- 


7 


4 


3 


4 


2 


4- 


8 


4 




4 


8 


4. 


9 


5 


5 


5 


5 


5. 


IC 


6 


1 


6 


1 


6. 


20 


12 


3 


12 


1 


12. 


30 


18 


5 


18 


2 


18. 


40 


24 


6 


24 


Q 


24. 


50 


30 


8 


30 


4 


30. 





35 


35 34 


6 


3.5 


3.5: 3 


7 


4 


1 


4 


1 4. 


8 


4 


7 


4 


6' 4. 


9 


5 


3 


5 


2j 5. 


10 


5 


9 


5 


8, 5. 


20 


11 


8 


11 


6 11. 


30 


17 




17 


5 17. 


40 


23 


6 


23 


3 23. 


50 


29 


6 


29 


li28- 






5 





4 





6 





5 





6 





6 





7 





7 





8 





7 


1 


6 


1 


5 


2 


5 


2 


2 


3 


3 


3 


^ 


4 


1 


3 


7I 



4 

0.4 
0.4 
0.5 



6 
6 
3 



12-6 
3 



P. P. 



70' 



20' 



TABLE VII.— LOGARITHMIC SINES, COSINES, 
AND COTANGENTS. 



TANGENTS. 



159' 





1 

2 

3 

j4 

5 
6 
7 
8 

10 

11 
12 
13 
11 
15 
16 
17 
18 
19_ 

20 

21 

22 

23 

21 

25 

23 

27 

28 

29_ 

30 

31 

32 

33 

34 

35 
36 
37 
38 

40 

41 
42 
43 
44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

M 

55 

58 

57 

53 

59 



Log. Sin. 



53 405 
53 440 
53 474 
53 509 
53 544 

53 578 
53 613 
53 647 
53 682 
53 716 



53 750 
53 785 
53 819 
53 854 
53_888 
53 922 
53 956 

53 990 

54 025 
54 059 



54 093 
54 127 
54 181 
54 195 
54 229 



54 283 
54 297 
54 331 
54 365 
54 398 



54 432 
54 466 
54 500 
54 534 
54 567 



54 601 
54 634 
54 668 
54 702 
54 735 



54 769 
54 802 
54 836 
54 869 
54 902 



54 936 

54 969 

55 002 
55 036 
55 069 



55 102 
55 135 
55 168 
55 202 
55 235 



55 268 
55 301 
55 334 
55 367 
55 400 



P5 433 



Log. Cos 



d. 

35 

34 
34 
35 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

33 
34 
34 
34 
33 
34 
34 
33 
34 
33 

33 
33 
34 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 



d. 



Log. Tan. 



56 106 
56 146 
56 185 
56 224 
56 263 



56 303 
56 342 
56 381 
56 420 
56 459 



56 498 
56 537 
56 576 
56 615 
56 654 



56 693 
56 732 
56 771 
56 810 
56 848 



56 887 
56 926 

56 965 

57 003 
57 042 



57 081 
57 119 
57 158 
57 196 
57 235 



57 274 
57 312 
57 350 
57 389 
57 427 



57 466 
57 504 
57 542 
57 581 
57 619 



657 
696 
734 
772 
810 



848 
886 
925 
963 
001 



9. 58 



039 
077 
115 
153 
190 

228 
266 
304 
342 
3^0 

417 



Log. Cot. 



c.d. 



39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
38 

39 
39 
39 
39 
38 

39 
38 
3? 
38 
38 

39 
38 
38 
38 
38 

39 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 

37 

38 
38 
38 
37 
38 

37 
c.d. 



Log. Cot. 



43 893 
43 854 
43 815 
43 775 
43 736 



43 697 
43 658 
43 619 
43 580 
43 540 



43 501 
43 462 
43 423 
43 384 
43 346 



43 307 
43 268 
43 229 
43 19G 
43 151 



43 112 
43 074 
43 035 
42 996 
42 958 



42 919 
42 880 
42 842 
42 803 
42 76c 



42 726 
42 687 
42 649 
42 611 
42 572 



42 534 
42 495 
42 457 
42 41G 
42 38C 



42 342 
42 304 
42 266 
42 227 
42 189 



42 151 
42 113 
42 075 
42 037 
41 999 



41 961 
41 923 
41 885 
41 847 
41 809 



41 771 
41 733 
41 695 
41 658 
41 620 



041 582 



Log. Tan 



Log. Cos. 



97 298 
97 294 
97 289 
97 285 
97 280 



97 275 
97 271 
97 266 
97 261 
97 257 



97 255 
97 248 
97 243 
97 238 
97 234 



97 229 
97 224 
97 220 
97 215 
97 210 



97 206 
97 201 
97 196 
97 191 
97 187 



97 182 
97 177 
97 173 
97 168 
97 163 



97 159 
97 154 
97 149 
97 144 
97 140 



97 135 
97 130 
97 125 
97 121 
97 116 



97 111 
97 106 
97 102 
97 097 
97 092 



97 087 
97 082 
97 078 
97 073 
97 068 



97 063 
97 058 
97 054 
97 049 
97 044 



97 039 
97 034 
97 029 
97 025 
97 020 



110' 



997 015 
Log. Sin. 

566 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
j46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
_31 

30 

29 
28 
27 
16 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 
11 



10 

9 
8 

7 
_6 

5 
4 
3 
2 
1 

O 



P. P. 





39 


39 


6 


3.9 


39 


7 


4 


6 


4 


5 


8 


5 


2 


5 


2 


9 


5 


9 


5 


8 


10 


6 


6 


6 


5 


20 


13 


1 


13 





30 


19 


7 


19 


5 


40 


26 


3 


26 





50 


32 


9 


32 


5 



38 38 37 



6 

7 

8 

9 

10 

20 

30 

40 

50 



3 


8 


3 


8 


3- 


4 


5 


4 


4 


4. 


5 


1 


5 





5 


5 


8 


5 


7 


5. 


6 


4 


6 


3 


6 


12 


8 


12 


6 


12. 


19 


2 


19 





18. 


25 


6 


25 


3 


25. 


32 


1 


31 


6 


31. 





35 


34 


34 


6 


35 


3.4 


3. 


7 


4.1 


4 





3. 


8 


4.6 


4 


6 


4. 


9 


5.2 


5 


2 


5. 


10 


5.8 


5 


7 


5. 


20 


11.6 


11 


5 


11- 


30 


17-5 


17 


2 


17. 


40 


23.3 


23 





22. 


50 


29.1 


28 


7 


28. 



33 



6 

7 

8 

9 

10 

20 

30 

40 

50 



3 


3 


3 


3 


9 


3. 


4 


4 


4 


5 





4 


5 


6 


5 


11 


1 


11 


16 


7 


16 


22 


3 


22 


27 


9 


27. 



33 

3 



5 4 



6 

7 

8 

9 

10 

20 

30 

40 

50 






5 


0. 





6 


0. 





6 


0. 





7 


0. 





8 


0. 


1 


6 


1. 


2 


5 


2. 


3 


3 


3. 


4 


1 


3- 



P. p. 



I 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



158' 



Log. Sin. I d. 



55 433, 
55 4661 
55 498 
55 5311 
55 564! 



55 597 
55 630 
55 662 
55 695 
55 728 



55 760 
55 793 
55 826 
55 858 
55 891 



55 923 
55 958 

55 988 

56 020 
56 05.^1 



56 085 
56 118 
56 150! 3I 



33 
32 
33 
33 

32 
33 
32 
33 
32 

32 
32 
33 
32 

32 

I 32 
j 32 

! 32 

! 32 

32 

32 
32 
32 



56 182 
56 214 



56 247 
56 279 
56 311 
56 343 
56 375 



56 407 
56 439 
56 471 
56 503 
56 535 



56 567 
56 599 
56 631 
56 663 
56 fiPn 



56 727 
56 758 
56 790 
56 822 
56 854 



56 885 
56 917 
56 949 

56 980 

57 012 



57 043 
57 075 
57 106 
57 138 
57 169 



57 201 
57 232 
57 263 
57 295 
57 326 
57 357 



Log. Cos. 



32 
32 
32 
32 
32 
32 

32 
32 
32 
32 
32 
32 
32 
32 
31 
32 

32 
3l 
32 
31 
32 

31 
31 
32 
31 
31 

3l 
31 
31 
3l 

31 

31 
31 
31 
3l 
3l 

'31 



Log. Tan 



58 417 
58 455 
58 493 
58 531 
58 56R 



58 606 
58 644 
58 681 
58 719 
58 756 



58 794 
58 831 
58 869 
58 906 
58 944 



58 981 

59 019 
59 056 
59 093 

59 T?^"" 



59 168 
59 205 
59 242 
59 280 
59 317 



59 354 
59 391 
59 428 
59 465 
59 50!^! 



59 540 
59 577 
59 614 
59 651 
59 688 



724 
761 
798 
835 
872 

909 
946 
982 
019 
056 

093 
129 
166 
203 

239 

276 
312 
349 
386 

422 

459 
495 
531 
568 
604 
641 



Log. Cot. c. d 



c.d. 

38 

37 , 

38 

37 

37 
38 
37 
37 
37 
37 
37 
37 

3Z 

37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

36 
37 
37 
37 
36 

37 
37 
36 
37 
36 

37 
36 
37 
36 
36 

36 
36 
37 
36 
35 

30 
36 
36 
36 
36 
36 



Log. Cot. 



0-41 582 
0-41 544 
041 507 
0-41 469 
0-41 431 



0.41 394 
0.41 356 
0.41 318 
0.41 281 
0.41 243 9 



Log. Cos 



• 97 015 

.97 010 

97 005 

97 000 

96 995 



0-41 206 
0-41 168 
041 131 
041 093 
041 056 



041 018 
040 981 
0.40 944 
0.40 906 
0-40 RSP 



040 832 
0-40 794 
0.40 757 
0-40 720 
0.40 683 



0.40 646 
0.40 608 
0.40 571 
0.40 534 
0.40 497 





40 460 

40 423 

0.40 386 

0.40 349 

040 312 



040 275 
0.40 238 
0.40 201 
0-40 164 
0.40 IQR 



040 091 
0.40 054 
040 017 
039 980 
0-39 944 



39 907 
39 870 
039 833 
039 797 
039 760 



39 724 
39 687 
0.39 650 
0.39 614 
0-39 577 



0.39 541 
0.39 5C/1 
0.39 468 
039 432 
0.39 395 



.96 991 
96 986 
96 981 
96 976 

.96 971 



96 966 
96 961 
96 956 
96 952 
96 947 



98 942 
96 937 
96 932 
96 927 
96 otoo 



96 917 
96 912 
96 907 
96 902 
96 897 



96 892 
96 887 
96 882 
96 877 
96 873 



96 868 
96 863 
96 858 
96 853 
96 848 



98 843 
96 838 
96 833 
96 828 
96 823 



96 818 
96 813 
96 808 
96 802 
96 797 



96 792 
96 787 
96 782 
96 777 
96 772 



96 767 
96 762 
96 757 
96 752 

or^ 74.7 



96 742 
96 737 
96 732 
96 727 
96 721 

96 71R 



d. 



r. Tan.jM.g. Sin. d. 
507 



60 

59 
58 
57 

55 
54 
53 
52 
51 



40 

39 
38 

37 
31 
35 
34 
33 
32 
31 



30 

29 
28 
27 
2§_ 

25 
24 
23 
22 
21 

30 

19 
18 
17 
16 



15 
14 
13 
12 
11 
10 
9 
8 



P. P. 





38 


37 


37 


6 


38 


3-7 


3.7 


7 


4 


4 


4 


A 


4 


3 


8 


5 





5 


c 


4 


9 


9 


5 


7 


5 


6 


5 


5 


10 


6 


3 


6 


2 


6 


1 


20 


12 


6 


12 




12 


3 


30 


19 


C 


18 


7 


18 


5 


40 


25 


3 


25 


C 


24 


6 


50 


31 


6 


31 


2 


30 


8 





36 


36 


6 


3-6 


36 


7 


4 


2 


4 


2 


8 


4 


8 


4 


8 


9 


5 


5 


5 


4 


10 


6 


1 


6 





20 


12 


1 


12 





30 


18 


2 


18 





4G 


24 


3 


24 





50 


30 


4 


30 








33 


32 


3 


6 


3.3 


3.2 


3. 


8 


3 


8 


3 


8 


3. 


7 


4 


4 


4 


3 


4. 


9 


4 


9 


4 


9 


4. 


10 


5 


5 


5 


4 


5. 


20 


11 





10 




10. 


30 


16 


5 


16 


2 


16 


40 


22 





21 


6 


21 


50 


27 


5 


27 


1 


26 



31 



6 

7 

8 

9 

10 

20 

30 

40 

50 



31 

31 





5 




> 


4' 


6 


0.5 


0.5 


0.4 


7 


0.6 





6 





5 


8 


0.7 





6 





6 


9 


0.8 





7 





7 


10 


0-9 





P 





7 


20 


l-H 


1 


e 


1 


f) 


30 


2.7 


2 


5 


2 


2 


40 


36 


3 


3 


3 





50 


46 


4 


1 


3 


7 



P. p. 



68' 



23' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



157° : 



Log. Sin. 



57 357 
57 389 
57 420 
57 451 
57 482 



57 513 
57 544 
57 576 
57 607 
57 638 



57 669 
57 700 
57 731 
57 762 
57 792 



57 823 
57 854 
57 885 
57 916 
57 947 



57 977 

58 008 
58 039 
58 070 
58 100 



58 131 
58 162 
58 192 
58 223 
58 253 



58 284 
58 314 
58 345 
58 375 
58 406 



58 436 
58 466 
58 497 
58 527 
58 557 



58 587 
58 618 
58 648 
58 678 
58 708 



58 738 
58 769 
58 799 
58 829 
58 859 



58 889 
58 919 
58 949 

58 979 

59 009 



59 038 
59 068 
59 098 
59 128 
59 158 



Q . '■>9 1 88 



Log. Cos. d 



d. 



31 
31 
31 
3l 

31 
31 
3l 
31 
31 

31 
31 
31 
31 
30 

31 
31 
31 
30 
31 

30 
31 
30 
31 
30 

30 
31 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

29 
30 
30 
29 
30 

30 



Log. Tan. 



60 641 
60 677 
60 713 
60 750 
60 786 



60 822 
60 859 
60 895 
60 931 
60 967 



61 003 
61 039 
61 076 
61 112 
61 148 



61 184 
61 220 
61 256 
61 292 
61 328 



61 364 
61 400 
61 436 
61 472 
61 507 



61 543 
61 579 
61 615 
61 651 
61 686 



61 722 
61 758 
61 794 
61 829 
61 865 



61 901 
61 936 

61 972 

62 007 
P.9 043 



62 078 
62 114 
62 149 
62 185 
62 220 



62 256 
62 29l 
62 327 
62 362 
62 397 



62 433 
62 468 
62 503 
62 539 
62 574 



62 609 
62 644 
62 679 
62 715 
62 750 



9. 62 785 



Log. Cot, 



c. d, 

36 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
35 

36 
36 
35 
36 
35 

36 
35 
36 
35 
35 
36 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 



Log. Cot. 



39 359 
39 322 
39 286 
39 250 
39 213 



39 177 
39 141 
39 105 
39 069 
39 032 



38 996 
38 960 
38 924 
38 888 
38 852 



38 816 
38 780 
38 744 
38 708 
38 672 



38 636 
38 6C0 
38 564 
38 528 
38 492 



Log. Cos. 



38 456 
38 420 
38 385 
38 349 
?8 313 



38 277 
38 242 
38 206 
38 170 
38 135 



38 098 
38 063 
38 028 
37 992 
37 957 



37 921 
37 886 
37 850 
37 815 
37 779 



37 744 
37 708 
37 673 
37 637 
37 602 



37 567 
37 531 
37 496 
37 461 
37 426 



37 390 
37 355 
37 320 
37 285 
37 250 



0.37 215 



Log. Tan. 



96 716 
96 7ll 
96 706 
96 701 
96 696 



96 691 
96 686 
96 681 
96 675 
96 670 



96 665 
96 660 
96 655 
96 650 
96 644 



96 639 
96 634 
96 629 
96 624 
PR 619 



96 613 
96 608 
96 603 
96 598 
96 593 



96 587 
96 582 
96 577 
96 572 
96 567 



96 561 
96 556 
96 551 
96 546 
96 540 



96 535 
96 530 
96 525 
96 519 
96 514 



96 509 
96 503 
96 498 
96 493 
96 488 



96 482 
96 477 
96 472 
96 466 
96 46l 



96 456 
96 450 
96 445 
96 440 
96 434 



96 429 
96 424 
96 418 
96 413 
96 408 



96 402 



111' 



Log. Sin, 
568 



d. 

5 
5 
5 
5 
5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
T 



60 

59 
58 

57 

55 
54 
53 
52 
-51 
50 
49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 

30 

29 
28 
27 
26 

25 
24 
23 
22 
21 

30 

19 
18 
17 
16 

15 
14 
13 
12 
11 

10 

9 
8 

7 
6 

5 
4 
3 

2 

1 
O 



P.P. 





36 


36 


6 


3.6 


36 


7 


4.2 


4 


2 


8 


4.8 


4 


8 


9 


5.5 


5 


4 


10 


6 1 


6 





20 


12.1 


12 





30 


18.2 


18 





40 


24.3 


24 





50 


30.4 


30 






i 



6 

7 

8 

9 

10 

20 

30 

40 

50 



31 

3l 



31 

31 



30_ 

30 
3.5 
4.0 
4.6 
5.1 
-,101 
30 15.2 
40120.3 
50)25.4 



6 
7 
8 
9 
10 
20 



30 

30 

35 

40 

4.5 

5-0 

10.0 

15-0 

20.0 

25.0 



29 

2-9 



6 
7 
8 
9 

10 
20 
30 
40 
50 



5 5 

0.5j0.5 
6!0 
710 
8 
9|0 
8!l 



63 
6]4 



P. P. 





35 


35 : 


6 


3-5 


3-5 


7 


4 


1 


4 


1 


8 


4 


7 


4 


6 ^1 


9 


5 


3 


5 


2 i 


10 


5 


9 


5 


00 


20,11 


g 


11 


6 


30 17 


7 


17 


5 I 


40,23 


6 


23 


00 


50.29 


6 


29 


1 



67* 



TABLE VII.— LOGARITHMIC SINES, COSINES, 
AND COTANGENTS. 



TANGENTS, 



156' 



Log. Sin, 



9.59 
59 



60 



188 
217 
247 
277 
306 

336 
366 
395 
425 
454 

484 
513 
543 
572 
602 

631 
681 
690 
719 
749 

778 
807 
837 
886 
895 

924 
953 
982 
012 
041 

070 
099 
128 
157 
186 

215 
244 
273 
301 
330 

359 
388 
417 
445 
474 

503 
532 
560 
589 
618 

646 
675 
703 
732 
760 
789 
817 
846 
874 
903 
931 



Log. Cos, d. 



d. 



29 
30 
29 
29 

30 

29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
28 
29 

29 
28 
29 
28 
29 

28 
29 
28 
28 
29 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 
28 



Log. Tan. c.d, 



62 785i 
62 820 
62 855 
62 890 
62 925 



62 960 

62 995 

63 030 
63 065 
63 100 



63 135 
63 170 
63 205 
63 240 
63 275 



63 310 
63 344 
63 379 
63 414 
63 449 



63 484 
63 518 
63 553 
63 588 

63 622 



63 657 
63 692 
63 726 
63 761 
63 795 



63 830 
63 864 
63 899 
63 933 
63 968 



64 002 
64 037 
64 071 
64 106 
64 140 



64 174 
64 209 
64 243 
64 277 
64 312 



64 346 
64 380 
64 415 
64 449 
64 483 



64 517 
64 551 
64 585 
64 620 
64 654 



64 688 
64 722 
64 756 
64 790 
64 824 



64 85R 



Log. Cot 



35 
35 
35 
35 
35 
35 
35 
35 
35 

35 
35 
35 
34 
35 

35 
34 
35 
35 
34 

35 

34 
34 
35 
34 
34 
35 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

3i 
34 
34 
34 
34 

34 

34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
c.d. 



Log. Cot. 



37 215 
37 179 
37 144 
37 109 
37 074 



37 039 
37 004 
36 969 
36 934 
36 899 



36 864 
36 829 
36 794 
36 76C 
36 725 



36 69C 
36 655 
36 620 
36 585 
36 551 



36 516 
36 481 
36 447 
36 412 
36 377 



36 343 
36 308 
36 211 
36 239 
36 204 



36 170 
36 135 
36 101 
36 066 
36 032 



35 997 
35 963 
35 928 
35 894 

35 859 



35 825 
35 791 
35 756 
35 72 
35 688 



35 653 
35 619 
35 585 
35 551 
35 517 



35 482 
35 448 
35 414 
35 38C 
35 346 



35 312 
35 278 
35 244 
35 209 
35 175 



n-35 141 
Log. Tan. 



Log. Cos. d. 



96 402 
96 397 
96 392 
96 386 
96 381 



96 375 
96 370 
96 365 
96 359 
96 354 



96 349 
96 343 
96 338 
96 332 
96 327 



96 321 
96 316 
96 311 
96 305 
96 300 



96 294 
96 289 
96 283 
96 278 
96 272 



96 267 
96 261 
96 256 
96 251 
86 245 



96 240 
96 234 
98 229 
96 223 
96 218 



96 212 
96 206 
96 201 
96 195 
96 390 



96 184 
96 179 
96 173 
86 168 
96 162 



96 157 
96 151 
96 146 
96 140 
96 134 



96 129 
96 123 
96 118 
96 112 
96 106 



96 101 
96 095 
96 090 
96 084 
96 078 



96 073 



Log. Sin, 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
M. 
45 
44 
43 
42 
41 

40 

39 

38 

37 

J6 

35 
34 
33 
32 
-31 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 

17 
16 

15 
14 
13 
12 
11 

10 

9 
8 
7 
6 
5 
4 
3 
2 
1 





P.P. 



35 

3.5 
4 



2C;ii 

30 17 
4023 
50129 



35 

3-5 



34 



6 

7 

8 

9 

10 

20 

30 

40 

50 



1 3 


4 


3. 


4 





3. 


4 


6 


4. 


5 


2 


5. 


5 


7 


5. 


11 


5 


11. 


17 


2 


17. 


23 





22. 


28 


7 


28. 



34 

4 
9 
5 
1 
6 
3 

6 
3 



6 


3. 


7 


3- 


8 


4. 


9 


4. 


10 


5. 


2C 


10. 


3C 


15. 


4C 


20. 


5G 


25. 



30 


5 

5 










29 


29 


28 


6 


2.9 


2-9 


2.8 


7 


3 


4 


3 


4 


3 


3 


8 


3 


9 


3 


8 


3 


8 


9 


4 


4 


4 


3 


4 


3 


10 


4 


9 


4 


8 


4 


7 


20 


8 


8 


9 


6 


9 


5 


30 


14 


7 


14 


5 


14 


2 


40 


19 


6 


19 


3 


19 





50 


24 


6 


24 


1 


23 


7 



6 5_ 

60-60.5 



7,0 
80 
90 
10:1 
202 
30 3 
40 4 
50 5 









OiO 



4.1 



P.P. 



113° 



569 



66= 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
34° AND COTANGENTS. 155» 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
14^ 

15 
16 
17 
18 
19 

20 

21 

22 

23 

21 

25 

26 

27 

28 

21 

30 

31 

32 

33 

3± 

35 
36 
37 
38 

iL 
40 

41 
42 
43 
44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 
56 
57 
58 
59^ 

60 



Log. Sin. 



60 931 
60 959 

60 988 

61 016 
61 044 



61 073 
61 101 
61 129 
61 157 
61 186 



61 214 
61 242 
61 270 
61 298 
61 326 



61 354 
61 382 
61 410 
61 438 
61 466 



61 494 
61 522 
61 550 
61 57? 
61 606 



61 634 
61 661 
61 689 
61 717 
61 745 



61 772 
61 800 
61 828 
61 856 
61 883 



61 911 
61 938 
61 966 

61 994 

62 021 



62 049 
62 076 
62 104 
62 131 
62 158 



62 186 
62 213 
62 241 
62 268 
62 295 



62 323 
62 350 
62 377 
62 404 
62 432 



62 459 
62 486 
62 513 
62 540 
62 567 



62 595 



Log. Cos. d 



28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
27 
28 
28 
28 
27 
28 
27 
28 

27 
28 
27 
28 
27 

27 
27 
27 
28 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 



Log. Tan, c. d. Log. Cot. Log. Cos 



64 858 
64 892 
64 926 
64 960 
64 994 



65 028 
65 062 
65 096 
65 129 
65 163 



65 197 
65 231 
65 265 
65 299 
65 332 



65 366 
65 400 
65 433 
65 467 
65 501 



65 535 
65 568 
65 602 
65 635 
65 669 



65 703 
65 736 
65 770 
65 803 
65 837 



65 870 
65 904 
65 937 

65 971 

66 004 



66 037 
66 071 
66 104 
66 137 
66 171 



66 204 
66 237 
66 271 
66 304 
66 337 



66 370 
66 404 
66 437 
66 470 
66 503 



66 536 
66 570 
66 603 
66 636 
66 669 



66 702 
66 735 
66 768 
66 801 
66 834 



9 
9 
9 
9 

9 66 867 

Log. Cot 



35 



972 
938 
904 
870 
836 



802 
769 
735 
701 
667 



633 
600 
566 
532 
499 



465 
431 
398 
364 
331 



129 
096 
062 
029 
996 

962 
929 
895 
862 
829 



795 
762 
729 
696 
^62 

629 
596 
563 

529 
496 



96 073 
96 067 
96 062 
96 056 
96 050 



96 045 
96 039 
96 033 
96 028 
96 022 



c.d. 



132 



Log. Tan, 



96 016 
96 011 
96 005 
95 999 
95 994 

95 988 
95 982 
95 977 
95 971 
95 965 



95 959 
95 954 
95 948 
95 942 
95 937 



95 931 
95 925 
95 919 
95 914 
95 908 



95 902 
95 896 
95 891 
95 885 
95 879 



95 873 
95 867 
95 862 
95 856 
95 850 



95 844 
95 838 
95 833 
95 827 
95 821 



95 815 
95 809 
95 804 
95 798 
95 792 



95 786 
G5 780 
95 774 
95 768 
95 763 



95 757 
95 751 
95 745 
95 739 
95 733 



9. 95 727 



114^ 



Log. Sin. 
570 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 
50 
49 
48 
47 
j46 

45 
44 
43 
42 

41 
40 

39 
38 
37 
_36 

35 
34 
33 
32 
_31 
30 
29 
28 
27 
26_ 

25 
24 
23 
22 
21 

30 

19 

18 

17 

16 

15 
14 
13 
12 
11 



10 

9 

8 

7 
_6 

5 
4 
3 
2 
_J_ 

O 



P. ? 





34 


33 


33 


6 


3.4 


3-3 


33 


7 


3 


9 


3 


9 


3 


8 


8 


4 


5 


4 


4 


4 


4 


9 


5 


1 


5 





4 


9 


10 


5 


6 


5 


6 


5 


5 


20 


11 


3 


11 


1 


11 





30 


17 





16 


7 


16 


5 


40 


22 


6 


22 


3 


22 





50 


28 


3 


27 


9 


27 


.5 

\ 

1 



37 



6 

7 

8 

9 

10 

20 

30 

40 

50 



2 


7 


2 


3 


2 


3 


3 


6 


3- 


4 


1 


4 


4 


6 


4. 


9 


1 


9 


13 


7 


13 


18 


3 


18. 


22 


9 


22. 



27 

7 
1 
6 

5 

5 

5 



6 5 
5 
6 
7 
8 
9 
8 
7 
6 
6 



6 





6,0 


7 





70. 


8 





80. 


9 





90 


IC 


1 


00. 


20 


2 


01. 


30 


3 


02. 


40 


4 


03. 


50 


5 


0,4. 



P. p. 





28 


28 


6 


2.8 


2.8 


7 


3 


3 


3 


2 


8 


3 


8 


3 


7 


9 


4 


3 


4 


2 


10 


4 


7 


4 


6 


20 


9 


5 


9 


3 


30 


14 


2 


14 





40 


19 





18 


6 


50 


23 


7 


23 


3 



65^ 



P5^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



154° 



Log. Sin.! d. 



62 595 
62 622 
62 649 
62 676 
62 703 



62 730 
62 757 
62 784 
62 811 
62 838 



62 864 
62 891 
62 918 
62 945 
62 972 



62 999 

63 025 
63 052 
63 079 
63 106 



63 132 
63 159 
63 186 
63 212 
63 239 



63 266 
63 292 
63 319 
63 345 
63 372 

63 398 
63 425 
63 451 
63 478 
63 504 



63 530 
63 557 
63 583 

63 609 
63 63 B 



63 662 
63 688 
63 715 
63 741 
63 767 



63 793 
63 819 
63 846 
63 872 
63 898 



63 924 
63 950 

63 976 

64 002 
64 028 



64 054 
64 080 
64 106 
64 132 
64 158 



9-64 184 
Log. Cos. 



27 
27 
27 
27 

27 
27 
27 
27 
27 
26 
27 
27 
27 
26 

27 
26 
27 
26 
27 

26 
27 
26 
26 
26 

27 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
28 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

25 



Log. Tan. c, d 



867 
900 
933 
966 
999 



032 
065 
097 
130 
163 



196 
229 
262 
294 
327 



360 
393 
425 
458 
491 



523 
556 
589 
621 
654 



687 
719 
752 
784 
817 



849 
882 
914 
947 
979 



012 
044 
077 
109 
14l 



174 
206 
238 
271 
303 



335 
368 
400 
432 
464 



497 
529 
561 
593 
625 

657 
690 
722 
754 
786 



68 818 
Log. Cot. 



32 
33 
33 
33 

33 
33 
32 
33 
33 

33 
32 
33 
32 
33 

32 
33 
32 
33 
32 

32 
33 
32 
32 
33 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 
32 
32 
32 
32 
32 

32 
32 
32 
32 
32 
32 

c. d. 



Log. Cot. 



33 132 
33 100 
33 067 
33 034 
S3 001 

32 968 
32 935 
32 902 
32 869 
32__816 

32 803 
32 771 
32 738 
32 705 
32 672 



32 640 
32 607 
32 574 
32 541 
32 509 



32 476 
32 443 
32 411 
32 378 
32 345 



313 
280 
248 
215 
183 



150 
118 
085 
053 
020 



31 988 
31 955 
31 923 
31 891 
31 858 



31 826 
31 792 
31 76l 
31 72? 
31 696 



31 664 
31 632 
31 600 
31 567 
31 535 



31 503 
31 471 
31 439 
31 406 
31 374 



31 342 
31 310 
31 278 
31 246 
31 214 



0-31 182 
Log, Tan. 



Log. Cos, 



95 727 
95 721 
95 716 
b5 710 
0(3 704 

95 698 
95 692 
95 686 
95 680 
95 674 



95 668 
95 662 
95 656 
95 650 
95 644 



95 638 
95 632 
95 627 
95 621 
95 615 



95 609 
95 603 
95 597 
95 591 
95 585 



95 579 
95 573 
95 567 
95 561 
95 555 



95 549 
95 543 
95 537 
95 530 
95 524 



95 518 
95 512 
95 506 
95 500 
95 494 



95 488 
95 482 
95 476 
95 470 
95 464 



95 458 
95 452 
95 445 
95 439 
95 433 



95 427 
95 42l 
95 415 
95 409 
95 403 



95 397 
95 390 
95 384 
95 378 
95 372 

9-95 366 
Log. Sin. 



60 

59 
58 
57 

li 

55 
54 
53 
52 
11 
50 
49 
48 
47 
j46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
_36 

35 
34 
33 
32 
11 
30 
29 
28 
27 
26 

25 

24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 

13 

12 

Jl 

10 

9 

8 

7 

6 

5 
4 
3 
2 
1 



P. P. 





33 


32 


32 


6 


33 


3.2 


3-2 


7 


3 


8 


3 


8 


3 


7 


8 


4 


4 


4 


3 


4 


2 


9 


4 


9 


4 


9 


4 


8 


IC 


5 


5 


5 


4 


5 


3 


20 


11 





10 


8 


10 


5 


30 


16 


5 


16 


2 


16 





40 


22 





21 


g 


21 


3 


50 


27 


5 


27 


1 


26 


6 



6 


2. 


7 


3- 


8 


3. 


9 


4. 


10 


4- 


20 


9. 


30 


13- 


40 


18. 


50 


22. 



27 

7 
1 
6 

5 

5 

5 





26 


26 


2 


6 


2-6 


2.6 


2. 


7 


3 


1 


3 





3. 


8 


3 


5 


3 


4 


3 


9 


4 





3 


9 


3 


10 


4 


4 


4 


3 


4. 


20 


8 


8 


8 


6 


8- 


30 


]3 


2 


13 





12. 


40 


17 


6 


17 


3 


17. 


50 


22 


1 


21 


6 


21. 





6 


6 


5 


6 


0-6 


0-6 


0.5 


7 





7 





7 


0-6 


8 





8 





8 


0.7 


9 


1 








9 


0.8 


10 


1 


1 


1 





0.9 


20 


2 


1 


2 





1.8 


3C 


3 


2 


3 





2.7 


4C 


4 


3 


4 





3.6 


50 


5 


4 


5 





4.6 



P. p. 



571 



64* 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
36° AND COTANGENTS. 153 



11 

3° \ 



/ 


Log. Sin. 





9-64 184 


1 


9 


.64 210 


2 


9 


• 64 236 


3 


9 


• 64 262 


4 


9 


• 64 287 


5 


9 


• 64 313 


6 


9 


.64 339 


7 


9 


64 365 


8 


9 


64 391 


9 


9 


64 416 


10 


9 


64 442 


11 


9 


64 468 


12 


9 


64 493 


13 


9 


64 519 


14 


9 


64 545 


15 


9 


64 570 


16 


9 


64 596 


17 


9 


64 622 


18 


9 


64 647 


19, 


L 


64 873 


20 


9 


64 698 


21 


9 


64 724 


22 


9 


64 74S 


23 


9 


64 775 


24 


9 


64 800 


25 


9 


64 828 


26 


9 


64 851 


27 


9 


64 876 


28 


9 


64 902 


29 


9 


64 927 


30 


9 


64 952 


31 


9 


64 978 


32 


9 


65 003 


33 


9 


65 028 


34 


9. 


65 054 


35 


9. 


65 079 


36 


9- 


65 104 


37 


9. 


65 129 


38 


9. 


65 155 


39 


9. 


65 180 


40 


9. 


65 205 


41 


9- 


85 230 


42 


9. 


65 255 


43 


9- 


65 280 


44 


9. 


65 305 


45 


9. 


65 331 


46 


9. 


65 358 


47 


9. 


65 381 


48 


9. 


65 408 


49 


9. 


65 431 


50 


9. 


65 456 


51 


9. 


65 481 


52 


9. 


65 506 


53 


9. 


65 530 


54 


9. 


65 555 


55 


9. 


65 580 


56 


9. 


65 605 


57 


9. 


65 630 


58 


9. 


65 855 


59 


9. 


85 880 


60 


9 


85 704 




Lc 


g. Cos. 



26 
26 
26 
25 

26 
26 
25 
26 
25 

26 
25 
25 
26 
25 

25 
25 
26 
25 
25 

21 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
24 
25 

25 
25 
24 
25 
25 
24 



Log. Tan 



9-68 818 
9^68 850 
9. 68 882 
9.68 914 
9.68 946 



9.69 773 
9.69 805 
9.69 837 
9-69 868 
9.69 900 



9.68 978 

9.69 010 
9.69 042 
9 .69 074 
9-69 106 



9- 69 138 
9.69 170 
9-69 202 
9-69 234 
9-69 265 



9-69 297 
9-69 329 
9-69 361 
69-393 
69 425 



9-69 456 
9-69 488 
69 520 
9-69 552 
9-69 583 



9-69 615 
69 647 
9-69 678 
9-69 7lQ 
9-69 742 



69 931 
9-69 963 
9-69 994 
9 . 70 02« 
9.70 058 



9.70 089 
9.70 121 
9.70 152 
9.70 183 
9.70 215 



9.70 246 
9.70 278 
9.70 309 
70 341 
9-70 372 



70 403 
70 435 
70 486 
70 497 
70 5^9 



70 560 
70 59l 
70 623 
70 654 
70 685 



9-70 716 
Log. Cot. 



c. d 

32 
32 
32 
32 

32 
32 
32 
31 
32 

32 

32 
32 
32 
31 

32 
32 
31 
32 
32 

31 
32 
3l 
32 
31 

32 
31 
31 
32 
31 

31 
32 
31 
31 
31 

31 
31 
31 

32 
31 

31 
3i 
31 
31 
3l 

31 
31 
3l 

3l 
31 

3l 
31 
31 
31 
3l 

3l 
31 
3l 
31 
3l 

31 
c. d. 



Log. Cot 



0-31 182 
0-31 150 
0-31 117 
0.31085 
0-31 G53 



0.31 021 
0.30 989 
0.30 957 
0.30 926 
0.30 894 



0.30 862 
0.30 830 
0.30 798 
0.30 766 
. 30 734 



30 702 
30 670 
0.30 639 
0.30 607 
0.30 575 



Log. Cos. d. 



9-95 366 
9-95 360 
9-95 353 
9.95 347 
9-95 341 



9.95 335 
9.95 329 
9.95 323 
9.95 316 
9-95 310 



9-95 304 
9-95 298 
9.95 292 
9.95 285 
9.95 279 



9.95 273 
9.95 267 
9.95 260 
9.95 254 
9.95 248 



30 543 
30 5ll 
0.30 480 
0.30 448 
0.30 416 



0.30 384 
0.30 353 
0.30 321 
0.30 289 
0.30 258 



0.30 226 
0.30 194 
0.30 163 
-30 13l 
6.30 100 



0.30 068 
0.30 037 
0.30 005 
0.29 973 
0.29 942 



9.95 147 
9.95 141 
9.95 135 
9-95 128 
9-95 12P 



0.29 910 
0.29 879 
0.29 847 
0.29 816 
0.29 785 



0.29 753 
0.29 722 
0-29 890 
0-29 659 
0.29 628 



0.29 596 
0.29 585 
0-29 533 
0-29 502 
- 29 471 



29 439 
29 408 
29 377 
29 346 
29 314 



0-29 283 



Log. Tan. 



9-95 242 
9-95 235 
9.95 229 
9.95 223 
9.95 217 



9-95 210 
9-95 204 
9.95 198 
9.95 191 
9-95 185 



95 179 
95 173 
9.95 166 
9.95 160 
9-95 154 



9.95 116 
9.95 109 
9-95 103 
9-95 097 
9-95 090 



9-95 084 
9-95 078 
9^95 071 
9.95 065 
9-95 058 



95 052 
95 046 
95 039 
95 033 
95 026 



95 020 
95 014 
95 007 
95 001 
94 994 



9-94 988 
Log. Sin. 



60 

59 
58 

57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 

32 

11. 
30 

29 
28 

27 

21 

25 

24 

23 

22 

20 

19 

18 

17 

16. 

15 

14 

13 

12 

11 



10 

9 
8 
7 
8 
5 
4 
3 
2 
_I 
O 



P.P. 













33 33 


6 


3-2 


3.2 


7 


3 


8 


3 


7 


8 


4 


3 


4 


2 


9 


4 


9 


4 


8 ; 


10 


5 


4 


5 


3 


20 


10 


8 


10 


6 


30 


16 


2 


16 





40 


21 


6 


21 


3 


50 


l27 


1 


26 


6 



31 31 



6 

7 

8 

9 

10 

20 

30 

40 

50 





36 


25 


25 


6 


2-6 


2.5 


2-5 


7 


3 





3 





2-9 


8 


3 


4 


3 


4 


3-3 


9 


3 


9 


3 


8 


37 


10 


4 


3 


4 


2 


4-1 


20 


8 


6 


8 


5 


8-3 


30 


13 





12 


7 


12-5 


40 


17 


3 


17 





16-6 


50 


21 


6 


21 


2 


20.6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



24 

2-4 



6_ 

0^6 





1 

1 

2 



6 

0.6 



P.P. 



3 


i 


3 


1 


3 


7 


3 


g 


4 


2 


4 


I 


4 


7 


4 


6 


5 


2 


5 


1 


10 


5 


10 


3 


15 


7 


15 


5 


21 





20 


6 


26 


^ 


25 


8 



1X6° 



572 



63° 



TABLE VII.— LOGARITHMIC SINES, COSINES, 
AND COTANGENTS. 



TANGENTS, 



153° 



Log. Sin. 



65 704 
65 729 
65 754 
65 779 
65 803 



65 828 
65 853 
65 878 
65 902 
65 927 



65 951 

65 976 

66 001 
66 025 
66 050 



66 074 
66 099 
66 123 
66 148 

66 172 



66 197 
66 221 
66 246 
66 270 
66 294 



66 319 
66 343 
66 367 
66 392 

66 416 



66 440 
66 465 
66 489 
66 513 
66 537 



66 581 
66 588 
66 610 
66 634 
66 65S 



66 88^ 
66 706 
66 730 
66 754 
66 778 



66 802 
66 826 
66 850 
66 874 
68 898 



66 922 
66 946 
66 970 

66 994 

67 018 



67 042 
67 066 
67 089 
67 113 
67 137 



9-67 181 
Log. Cos, 



25 
24 
25 
24 

25 
24 
25 
24 
24 

24 
25 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
23 
24 

24 
24 
23 
24 
23 
24 



Log. Tan 



70 716 
70 748 
70 779 
70 810 
70 841 



70 872 
70 903 
70 935 
70 966 
70 997 



71 028 
71 059 
71 090 
71 121 
71 152 



71 183 
71 214 
71 245 
71 276 
71 307 



c. d, 



71 338 
71 369 
71 400 
71 431 
71 462 

71493 
71 524 
71 555 
71 586 
71 617 



71 647 
71 678 
71 709 
71 740 
71 771 



71 801 
71 832 
71 863 
71 894 
71 925 



71 955 

71 986 

72 017 
72 047 
72 078 



72 109 
72 139 
72 170 
72 201 
72 231 



72 262 
72 292 
72 323 
72 354 
72 384 



72 415 
72 445 
72 476 
72 506 
72 537 



72 567 
Log. Cot' 



31 
31 
31 
31 

31 
31 
3l 
31 
31 
31 
3l 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
30 
31 
31 
31 

30 
31 
31 
30 
31 

30 
31 
31 
30 
31 

30 
30 
31 
30 
31 

30 
30 
30 
31 
30 

30 
30 
30 
31 
30 

30 
30 
30 
30 
30 
30 



Logr Cot 



29 283 
29 252 
29 221 
29 190 
29 158 



29 127 
29 096 
29 065 
29 034 
29 003 



28 972 
28 940 
28 909 
28 878 
28 847 



28 816 
28 785 
28 754 

28 723 
28 692 



28 661 
28 630 
28 599 
28 568 

28 537 



Log. Cos 



28 506 
28 476 
28 445 
28 414 
28 383 

28 352 
28 321 
28 290 
28 260 
28 229 



28 198 
28 167 
28 136 
28 106 
28 075 



28 044 
28 014 
27 983 
27 952 
27 921 



27 891 
27 880 
27 830 
27 799 
27 768 



27 738 
27 707 
27 677 
27 646 
27 615 



27 585 
27 554 
27 524 
27 493 
27 463 



27 432 
I Log. Tan. 



94 988 
94 981 
94 975 
94 969 
94 962 



94 956 
94 949 
94 943 
94 936 
94 930 



94 923 
94 917 
94 910 
94 904 
94 897 



94 891 
94 884 
94 878 
94 871 
94 865 



94 858 
94 852 
94 845 
94 839 
94 832 



94 825 
94 819 
94 812 
94 806 
94 799 



94 793 
94 786 
94 779 
94 773 
94 766 



94 760 
94 753 
94 746 
94 740 
94 733 



94 727 
94 720 
94 713 
94 707 
94 700 



94 693 
94 687 
94 680 
94 674 
94 667 



94 660 
94 654 
9.4 647 
94 640 
94 633 



94 627 
94 620 
94 613 
94 607 
94 600 



94 593 
Log. Sin. 



60 

59 
58 
57 

55 
54 
53 
52 

50 

49 
48 
47 
48 



40 

39 
38 
37 
-36 
35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21 
20 
19 
18 
17 
li 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6 

5 
4 
3 
2 





P. P. 





31 




31 


30 


6 


3.1 


3.1 


30 


7 


3 


7 


3 


6 


3 


5 


8 


4 


2 


4 


1 


4 





9 


4 


7 


4 


g 


4 


6 


10 


5 


2 


5 


1 


5 


1 


20 


10 


5 


10 


3 


10 


\ 


30 


15 


7 


15 


5 


15 


2 


40 


21 





20 


6 


20 


3 


50 


26 


2 


25 


8 


25 


4 



6 


2 


7 


2 


8 


3 


9 


3 


10 


4 


20 


8 


30 


12 


40 


16 



50120 



25 

5 
9 

3 
7 
I 
3 
5 
6 
8 





34 34 


33 


6 


2-4 2-4 


2.3 


7 


2-8 2 


8 


2 


7 


8 


3.2| 3 


2 


3 


1 


9 


3-7 3 


6 


3 


5 


10 


4.1 4 





3 


§ 


20 


81 8 





7 


g 


30 


12.2!l2 





11 


7 


40 


16-3 16 





15 


f) 


50 


20.420 





19 


6 



7 6 6 



7 

8 

9 
10 
2012 
3013 
40:4 
50 5 



7:0 

8'0 
9 
01 
11 
3 2 
53 
6|4 
8l5 



610 
ZiO 
80 
00 
ill 
li2 
2|3 
3!4 
415 



P. P. 



573 



63° 



28' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



151' 



Log. Sin, 



67 161 
67 184 
67 208 
67 232 
67 256 



67 279 
67 303 
67 327 
67 350 
67 374 



67 397 
67 421 
67 445 
67 468 
67 492 



67 515 
67 539 
67 562 
67 586 
67 609 



67 633 
67 656 
67 679 
67 703 
67 726 



67 750 
67 773 
67 796 
67 819 
67 843 



67 866 
67 889 
67 913 
67 936 
67 959 



67 982 

68 005 
68 029 
68 052 
68 075 



68 098 
68 121 
68 144 
68 167 
68 190 



68 213 
68 236 
68 259 
68 282 
68 305 



68 328 
68 351 
68 374 
68 397 
68 420 



68 443 
68 466 
68 488 
68 511 
68 534 



68 557 
Log, Cos, 



23 

24: 

23 
24 

23 
2'3 
24 
23 
2'3 

23 
24 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 

23 
23 
23 
23 

23 

23 
23 
22 
23 

23 
23 
22 
23 
23 
22 



Log. Tan, 



c. d. 



72 567 
72 598 
72 628 
72 659 
72 689 



72 719 
72 750 
72 780 
72 811 

72 841 



72 871 
72 902 
72 932 
72 962 
72 993 



73 023 
73 053 
73 084 
73 114 
73 144 



73 174 
73 205 
73 235 
73 265 
73 295 



73 325 
73 356 
73 386 
73 416 
73 446 



73 476 
73 506 
73 536 
73 567 
73 597 



73 627 
73 657 
73 687 
73 717 
73 747 



73 777 
73 807 
73 837 
73 867 
73 897 



73 927 
73 957 

73 987 

74 017 
74 047 



74 076 
74 106 
74 136 
74 166 
74 196 



74 226 
74 256 
74 286 
74 315 
74 345 



9-74 375 
Log. Cot. 



Log. Cot. 



30 
30 
30 
30 

30 
30 
30 I 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
SO 

29 
30 
30 
30 
29 

30 
30 
30 
29 
30 

29 
c. d, 



0-27 432 
0.27 402 
0-27 37l 
0-27 341 
0.27 311 



Log. 



Cos. 



d. 



0.27 280 
0-27 25C 
0-27 219 
0-27 189 
0-27 159 



0-27 
0-27 
0-27 
0-27 
0-27 



94 593 
94 587 
94 580 
94 573 
94 566 



94 560 
94 553 
94 546 
94 539 
94 533 



0-26 
0.26 
0.26 
0.26 
0.?6 



128 9 

098 9 

067 

037 

007 

976 

946 

916 

886 

855 



0-26 825 
0.26 795 
0.26 765 
0.26 734 
0.26 704 



26 674 
26 644 
26 614 
26 584 
26 553 



26 523 
26 493 
26 463 
26 433 
26 403 



0.26 373 
0.26 343 
0.26 31S 
026 28S 

0.?B 253 



26 223 
26 183 
26 163 
26 133 
26 103 



0.26 
0.26 
0.26 
0.25 
0.25 



0-25 
0.25 
0.25 
0.25 
0.25 



073 
043 
013 
983 
953 

92? 

89c: 

863 
833 
804 



94 526 
94 519 
94 512 
94 506 
94 499 



94 492 
94 485 
94 478 
94 472 
94 465 



9-94 458 
9.94 451 
9 . 94 444 
9.94 437 
9.94 431 



9 . 94 424 
9.94 417 
94 410 
9 . 94 403 
9 94 396 
9.94 390 
9. 94 383 
9. 94 376 
9.94 369 
9. 94 362 



94 355 
94 348 
94 341 
94 335 
94 328 



94 321 
94 314 
94 307 
94 300 
94 293 



94 286 
94 279 
94 272 
94 265 
94 258 



0.25 774 
0.25 744 
0.25 714 
0.25 684 
0.25 654 



94 251 
94 245 
94 238 
94 231 
94 224 



0.25 625 
Log, Tan. 



9.94 217 
9.94 210 
9.94 203 
9 94 196 
9.94 189 



9-94 182 
Log. Sin. 



60 

59 
58 
57 
56 



50 
49 
48 
47 
46 



45 
44 
43 
42 
41_ 

40 

39 
38 
37 
36 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21 

SO 

19 
18 
17 
11 
15 
14 
13 
12 
J_l 

10 

9 

8 

7 

__6^ 

5 
4 
3 

2 

1 



P. P. 



30 

3.0 



10 



3 

5 3 
4 

6| 4 
l' 5 
110. 
215-0 
3120.0 
4125.0 



29_ 

2.9 
4 
9 
4 
9 
8 
7 
6 
6 





24 


6 


2.4 


7 


2.8 


8 


3.2 


9 


3.6 


10 


4.0 


20 


8.0 


30 


12.0 


40 


16.0 


50 


20.0 





33 


23 


2*^ 


6 


2.3 


2.3 


2. 


7 


2 


7 


2.7 


2. 


8 


3 


1 


3.0 


3. 


9 


3 


5 


3.4 


3. 


10 


3 


9 


38 


3. 


20 


7 


8 


7.6 


7. 


30 


11 


7 


11-5 


11. 


40 


15 


6 


15-3 


15. 


50 


19 


6 


19.1 


18. 





!• 
4 


f 


6 


6 


0.7 


0.6 


7 





8 


0-7 


8 





9 


0.8 


9 


1 





1.0 


10 


1 


1 


1. 1 


20 


2 


3 


2.1 


30 


3 


5 


32 


40 


4 


6 


4.3 


50 


5 


8 


5.4 



P. p. 



118' 



574 



61' 



39' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



150' 



Log. Sin. d. Log. Tan. c. d. Log. Cot 



68 557 
68 580 
68 602 
68 625 
68 648 



68 671 
68 693 
68 716 
68 739 
68 761 



68 784 
68 807 
68 829 
68 852 

68 87' 



68 89/ 
68 923 
68 942 
68 985 
68 987 



69 010 
69 032 
69 055 
69 077 
69 099 



69 122 
69 144 
69 167 
69 189 
69 21 T 



69 234 
69 253 
69 278 
69 301 
69 3?^ 



69 34o 
69 307 
69 390 
69 412 
69 434 



69 456 
69 478 
69 500 
69 523 
69 545 



69 567 
69 589 
69 611 
69 633 
69 655 



69 677 
69 699 
69 721 
69 743 
69 765 



69 787 
69 809 
69 831 
69 853 
69 875 



9.96 897 



23 
22 
23 
22 

23 
22 
23 
22 
22 

23 
22 
22 
22 
22 

22 
23 
22 
22 
22 

22 
22 
22 
22 
22 
22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 
22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
21 
22 
22 



9-74 
9-74 
9.74 
9.74 
9.74 



9.74 
9.74 
9-74 
9.74 
9-74 



9.74 
9-74 
9.74 
9.74 
9-74 



9.74 
9.74 

9.74 
9.74 
9.74 



375 
405 
435 
464 
494 

524 
554 
583 
613 
643 

672 
702 
732 
761 
791 

821 
850 
880 
909 
939 



9.74 
9-74 
9.75 
9.75 
9.75 



9.75 
9.75 
9.75 
9.75 
9.75 



998 
028 
057 
087 

116 
146 
175 
205 

234 



9.75 
9.75 
9.75 
9.75 
9.75 



9-75 
9.75 
9.75 
9.75 
9.75 



264 
293 
323 
352 
382 

411 
441 
470 
499 
529 



9.75 
9-75 
9.75 
9.75 
9.75 



9.75 
9.75 
9.75 
9.75 
9.75 



558 
588 
617 
646 
676 

705 
734 
764 
793 

822 



Log. Cos, d« Log. Cot 



9.75 
9.75 
9.75 
9.75 
9.75 



851 
881 
910 
939 
968 



9.75 
9.76 
9.76 
9-76 
9.76 



998 

027 
056 
085 
115 



9-76 144 



30 
30 
29 
30 

29 
30 
29 
29 
30 

29 
30 
29 
29 
30 

29 
29 
29 
29 
30 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29. 
29 
29 
29 

29 

29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 
29 



0.25 625 
0.25 595 
0-25 565 
0-25 53'5 
0^25^505 

0.25 476 
0.25 446 
0.25 416 
0.25 387 
Q_l2_5_357 

0.25 327 
0-25 297 
0.25 268 
0.25 238 
0.25 208 



9 
9 
9 
9 
9 

25 179 9 
9 





0.25 149 
0.25 120 
0.25 090 
0.25 030 



Log. Cos, 



0.25 031 
0.25 001 
0.24 972 
0.24 942 
0-24 913 



0.24 883 
0.24 854 
0.24 824 
0.24 795 
0.24 765 



0.24 736 
0.24 706 
0-24 677 
0-24 647 
0.24 61 



24 588 
24 559 
24 529 
24 500 
24 471 



0.24 441 
0.24 412 
0.24 383 
0.24 353 
0.24 324 



0.24 
0.24 
0.24 
0.24 
0.24 



0-24 
0.24 
0.24 
0.24 
0.24 



295 
265 
236 
207 
177 
14§ 
119 
090 
060 
031 



0-24 002 
0.23 973 
0-23 943 
0.23 914 
0.23 885 



0-23 856 
Log. Tan. 



94 182 
94 175 
94 168 
94 161 
94 154 



94 147 
94 140 
94 133 
94 126 
94118 



94 111 
94 104 
94 017 
94 090 
94 083 



94 076 
94 06^ 
94 062 
94 055 
94 048 



94 041 
94 034 
94 026 
94 019 
94 012 

94 005 
93 998 
93 991 
93 984 
.93_9_77 
93 969 
93 962 
93 955 
93 948 
93 941 



93 934 
93 926 
93 919 
93 912 
93 905 



93 898 
93 891 
93 883 
93 876 
93 869 



93 862 
93 854 
93 847 

93 S4C 
93 8?3 



9S 82v; 
93 818 
93 811 
93 804 
93 796 



93 789 
93 782 
93 775 
93 767 
93 760 



93 753 



119' 



Log. Sin. 
575 



60 

59 
58 
57 
5^ 

55 
54 
53 
52 
51 

50 

49 
48 

47 
j46 

45 
44 
43 
42 
41 

40 

39 
38 

37 
36 

35 
34 
33 
32 
31 

30 

29 
28 

27 
26 

25 
24 
23 
22 
21 

•;o 

19 
18 
17 
16 

15 
14 
13 
12 
11 



P. P. 





30 


29 


21 


6 


3-0 


2-9 


2. 


7 


3 


5 


3 


4 


3. 


8 


4 





3 


9 


3. 


9 


4 


5 


4 


4 


4. 


10 


5 


C 


4 


9 


4- 


20 


10 





9 


8 


9. 


30 


15 





14 




14. 


40 


20 


G 


19 


(3 


19. 


50 


25 





24 


6 


24. 



6 


2 


3 


7 


2 


7 


8 


3 





9 


3 


4 


10 


3 


8 


20 


7 


6 


30 


11 


5 


40 


15 


3 


50 


19 


1 





22 


22 


o 


6 


2.2 


2-2 


2 


7 


2 


6 


2 


5 


2 


8 


3 





2 


9 


2 


9 


3 


4 


3 


3 


3- 


10 


3 


7 


3 


6 


3- 


20 


7 


5 


7 


3 


7- 


30 


11 


2 


11 





10- 


40 


15 





14 


Q 


14 


50 


18 


7 


18 


3 


17- 






7 


0. 





9 




1 





L* . 


1 


1 


1. 


I 


2 


1- 


2 


5 


2. 


3 


7 


3. 


5 





4. 


le 


2 


5. 



7 
7 
8 
9 

1 
3 
5 
6 
8 



P. P. 



60' 



30* 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



149^ 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 



Log. Sin. 



69 897 
69 919 
69 940 
69 962 
69 984 



70 006 
70 028 
70 050 
70 071 
70 093 



70 115 
70 137 
70 158 
70 180 
70 202 



9-70 223 
9-70 245 
9-70 267 
9-70 288 
70 310 



20 

21 

22 

23 

24. 

25 

26 

27 

28 

29 

30 

31 
32 
33 
34 

35 
36 
37 
38 

39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



70 331 
9-70 353 
9-70 375 
9-70 396 
9-70 418 



70 439 
70 461 
70 482 
70 504 
70 525 



9 . 70 547 
9-70 568 
9-70 590 
9-70 611 
9-70 632 



70 654 
70 675 
70 696 
70 718 

70 739 



70 760 
70 782 
70 803 
70 824 
70 846 



70 867 
70 888 
70 909 
70 935 
70 952 



70 973 

70 994 

71 015 
71 036 
71 057 



71 078 
71 099 
9-71 121 
9.71 142 
9-71 163 



9.71 184 



Log, Cos 



22 
21 
22 
22 

21 
22 
22 
2l 
22 

2l 

22 
21 
21 
22 

2l 
2l 
22 
21 
21 

2l 
22 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
2l 
21 
21 

2l 
2l 
21 
21 
21 

21 
21 
21 
2l 
2l 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
2l 
21 
21 

21 



Log, Tan, 



76 144 
76 173 
76 202 
76 231 
76 260 



76 289 
76 319 
76 348 
76 377 
76 406 



76 435 
76 464 
76 493 
76 522 
76 551 



76 580 
76 609 
76 638 

76 667 
76 696 



76 725 
76 754 
76 783 
76 812 
76 841 



76 870 
76 899 
76 928 
76 957 
76 986 



77 015 
77 043 
77 072 
77 lOl 
77 130 



77 159 
77 188 
77 217 
77 245 
77 274 



77 303 
77 332 
77 361 
77 389 
77 418 



77 447 
77 476 
77 504 
77 533 
77 562 



77 591 
77 619 
77 648 
77 677 
77 705 



77 734 
77 763 
77 791 
77 820 
77 849 



77 877 



C.d. 

29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
28 
29 
29 
29 

29 
28 
29 
29 
29 

28 
29 
29 
28 
29 

29 
28 
29 
28 
29 

28 
29 
28 
29 
28 

29 
25 
28 
29 
28 

28 
29 
2g 
28 
29 

28 



Log. Cot 



Log. Cot.lLog. Cos, 



23 856 9 



23 827 
0.23 797 
0.23 768 
0-23 739 



0-23 710 
0.23 681 
0-23 652 
0.23 623 
0-23 594 



0.23 565 
0-23 535 
0.23 506 
0-23 477 
. 23 448 



0-23 419 
0-23 390 
0-23 361 
0-23 332 
0-23 303 



0-23 274 
0-23 245 
0-23 216 
0-23 187 



93 753 
93 746 
93 738 
93 731 
93 724 



93 716 
93 709 
93 702 
93 694 
93 687 



93 680 
93 672 
93 665 
93 658 
93 650 



93 643 
93 635 
93 628 
93 621 
93 613 



0.23 158 993 576 



0-23 129 
0.23 101 
0.23 072 
0.23 043 
0.23 014 



0-22 985 
0.22 956 
0.22 927 
0.22 898 
0-22 869 



0-22 841 
0.22 812 
0.22 783 
0-22 754 
0-22 725 



22 696 
22 668 
22 639 
22 610 
22 581 



0-22 553 
0.22 524 
0.22 495 
0.22 466 
0-22 438 



0.22 409 
0.22 380 
0.22 352 
0.22 323 
0-22 294 



0-22 266 
0.22 237 
0-22 208 
0-22 180 
22 151 



0.22 122 



cdilLog, Tan, 



9-93 606 
9-93 599 
9-93 591 
93 584 



93 569 
93 562 
93 554 
93 547 
93 539 



93 532 
93 524 
93 517 
93 509 
93 502 



93 495 
93 487 
93 480 
93 272 
93 465 



93 457 
93 450 
93 442 
93 435 
93 427 



9-93 420 
9-93 412 
9.93 405 
9.93 397 
9.. 93 390 



9-93 382 
9-93 374 
9-93 367 
9-93 359 
9.93 3? 



93 344 
93 337 
93 329 
93 321 
93 314 



9. 93 306 



130^ 



Log. Sin 
576 



d. 



60 

59 
58 
57 
56. 
55 
54 
53 
52 
51 



50 

49 
48 

47 
46 

45 
44 
43 
42 
41_ 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 

30 

29 
28 
27 
2§_ 

25 
24 
23 
22 
21 

20 

19 
18 

17 
JA 
15 
14 
13 
12 
11 

10 



P.P. 



22 

2-2 

2 

2 

3 

3 

7 
11 
14 
18 



21 

2-11 



21 

2-1 





8 


7 


•* 
i 


6 


0-8 


0-7 


0- 


7 


0-9 


0-9 


0- 


8 


1-0 


1.0 


0- 


9 


1.2 


1.1 


1- 


10 


1.3 


J. • 2 


1- 


20 


2-6 


2-5 


2- 


30 


4-C 


3-7 


3- 


40 


5-3 


5-0 


4. 


50 


6.6 


6-2 


5. 



P.P. 













29 29 28 


6 


2-9 


2-9 


2.8 


7 


3 


4 


3-4 


3.3 


8 


3 


9 


3-8 


3-8 


9 


4 


4 


4-3 


4.3 


10 


4 


9 


4-8 


4-7 


20 


9 


8 


9-6 


9-5 


3C 


14 


7 


14-5 


14-2 


40 


19 


6 


19.3 


19-0 


50 


24 


.6 


24.1 


23-7 



59= 



31^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



148' 



Log. Sin. 



9.71 
71 



184 
205 
226 
247 
268 



289 

310 
331 
351 
372 



393 
414 
435 
456 
477 



498 
518 

539 
560 
53^ 



d. 



601 
622 
643 
684 
684 



705 
726 
746 
767 
788 



808 
829 
849 
870 
891 

911 
932 
952 
973 
993 



014 
034 
055 
075 
098 

116 
136 
157 
177 
198 

218 
238 
259 
279 
299 



72 



319 
340 
360 
380 
400 

421 



21 
21 
21 
21 

21 
21 
21 
20 
21 

21 
21 
20 
21 
21 

21 
20 
21 
20 
21 

20 
21 
20 
21 
20 

21 
20 
20 
21 
20 

20 
20 
20 
21 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 



Log. Cos. d. 



Log, Tan 



77 877 
77 906 
77 934 
77 963 
77 992 



78 020 
78 049 
78 077 
78 106 
78 134 



78 163 
78 191 
78 220 
78 248 
78 277 



78 305 
78 334 
78 362 
78 391 
78 419 



78 448 
78 476 
78 505 
78 533 
78 561 



78 rgo 
78 618 
78 647 
78 675 
78 703 



78 732 
78 780 
78 788 
78 817 
78 845 



78 873 
78 902 
78 930 
78 958 
78 9R7 



79 015 
79 043 
79 071 
79 100 
79 128 



79 156 
79 184 
79 213 
79 241 
79 269 



c.d. 



79 297 
79 325 
79 354 
79 382 
79 410 



79 438 
79 466 
79 494 
79 522 
79 551 



79 579 



Log. Cot. 



28 
28 
28 
29 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 

28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 

28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 
28 

cTd! 



Log. Cot 



0.22 122 
0-22 094 
0.22 065 
0.22 037 
0.22 008 



0.21 
0-21 
0.21 
0.21 
0.21 



979 
951 
922 
894 
865 



0-21 
0.21 
0.21 
0.21 
0.21 



0.21 
0.21 
0.21 
0.21 
0.21 



837 
808 
780 
751 
723 
694 
666 
637 
609 
580 



0.21 552 
0.21 523 
0.21 495 
0.21 467 
0.21 438 



0.21 410 
0.21 381 
0.21 353 
0.21 325 
0.21,296 



0.21 268 
0.21 239 
0.21 211 
0.21 183 
0-21 154 



0.21 126 
0.21 098 
0.21 070 
0.21 041 
0.21 013 



0-20 985 
0.20 956 
0.20 9 
0.20 900 
0.20 872 



20 843 
20 815 
0.20 787 
0.20 759 
020 731 



20 702 
20 674 
20 646 
20 618 
20 590 



0.20 561 
020 533 
0.20 505 
0-20 477 
0.20 449 

. 20 421 



Log. Cos. 



Log. Tan. 



93 306 
93 299 
93 291 
93 284 
93 276 

93 268 
93 261 
93 253 
93 245 
93 238 



93 230 
93 223 
93 215 
93 207 
93 200 



93 192 
93 184 
93 177 
93 169 
93 161 



93 153 
93 146 
93 138 
93 130 
93 123 



93 115 
93 107 
93 100 
93 092 
93 084 



93 076 
93 069 
93 061 
93 053 
93 045 



93 038 
93 030 
93 022 
93 014 
93 006 



92 999 
92 991 
92 983 
92 975 
92 967 



92 960 
92 952j 
92 944| 
92 936. 
92J28J 
92 920! 
92 913' 
92 905 
92 897' 
92 889 



92 881 
92 873 
92 865 
92 853 
92 850 



9-92 842 



Log. Sin. 

577 



60 

59 
58 
57 

16 
55 
54 
53 
52 

_51 

50 

49 
48 
47 
j46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 

-31 
30 

29 
28 

27 
26^ 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 
11 

10 

9 
8 
7 
6 

5 
4 
3 

2 

1 



P. P. 





2* 


d 


2S 


2? 


6 


2.9 


2.8 


2. 


7 


3 


4 


3 


3 


3. 


8 


3 


8 


3 


8 


3- 


9 


4 


3 


4 


3 


4. 


10 


4 


8 


4 


7 


4. 


20 


9 


6 


9 


5 


9. 


30 


14 


5 


14 


2 


14. 


40 


19 


3 


19 





18. 


50 


24 


1 


23 


7 


23. 





21 


20 


2( 


6 


2.1 


2.0 


2. 


7 


2 


4 


2 


4 


2. 


8 


2 


8 


2 


7 


2. 


9 


3 


1 


3 


1 


3. 


10 


3 


5 


3 


4 


3. 


20 


7 





6 


8 


6. 


30 


10 


5 


10 


2 


10. 


40 


14 





13 


6 


13. 


50 


17 


5 


17 


1 


16. 






8 





9 


1 





1 


2 


1 


3 


2 


6 


4 





5 


3 


6 


6 



0^9 
0.9 
1.0 
1.1 
1.2 
2.5 
3.7 
5.0 
6.2 



P. P. 



68^ 



33° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



147' 



Log. Sin, 



o 

1 

c 

3 

±^ 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
li 
15 
16 
17 
18 

il 
20 

21 
22 
23 
24 

25 
26 
27 
28 
29, 
30 
31 
32 
33 
34 

35 

36 

37 

38 

31 

40 

41 

42 

43 

44 

45 

46 

47 

48 

ii 

50 

51 

!>■"; 

Ii4 

51 
5- 
57 
5ft 
5^ 



72 421 
72 441 
.2 461 
72 481 
72 501 

72 522 
72 542 
72 562 
72 582 
72 602 



72 622 
72 642 
72 662 
72 682 
72 702 



72 723 
72 743 
72 763 
72 783 
72 802 



72 822 
72 842 
72 862 
72 882 
72 902 



72 922 
72 942 
72 962 

72 982 

73 002 



73 021 
73 041 
73 06l 
73 081 
73 101 



73 120 
73 140 
73 160 
73 180 
73 199 



73 219 
73 239 
73 258 
73 278 
73 298 



73 317 
73 337 
73 357 
73 376 
73 396 



73 415 
73 435 
73 455 
73 474 
73 494 



73 513 
73 533 
73 552 
73 572 
73 591 



73 611 



Log. Cos. 



20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
19 

20 
20 
20 
20 
20 

20 
19 
20 
20 
20 

19 
20 
20 
19 
20 

19 
20 
19 
20 
19 

20 
19 
19 
20 
19 

19 
20 
19 
19 
19 

19 
20 
19 
19 
19 

19 
19 
19 
19 
19 

19 



Log. Tan 



579 
607 
635 
663 
691 

719 
747 
775 
803 
831 



859 
887 
915 
943 
971 

999 
027 
055 
083 
111 



139 
167 
195 
223 
251 

279 
307 
335 
363 
391 



418 
446 
474 
502 
530 



558 
586 

613 
641 
RR9 



697 
725 
752 
780 
808 



836 
864 
891 
919 
947 



975 
002 
030 
058 
085 



81 



141 
168 
196 
224 

251 



Log. Cot. 



c. d, 

28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

27 
28 
28 
28 
28 

27 
28 
28 
28 
27 

28 
28 
27 
28 
28 

27 
28 
27 
28 
28 

27 
28 
27 
28 
27 

28 

27 
27 
28 
27 
28 
27 
27 
28 
27 
27 

c. d. 



Log. Cot 



20 421 
20 393 
20 365 
20 337 
20 308 



20 280 
20 252 
20 224 
20 196 
20 168 



20 140 
20 112 
20 084 
20 056 
20 028 



20 000 
19 972 
19 944 
19 916 
19 888 



19 860 
19 832 
19 804 
19 776 
19 748 



19 721 
19 693 
19 665 
19 637 
19 609 



19 581 
19 553 
19 525 
19 497 
19 470 



19 442 
19 414 
19 386 
19 358 
19 330 



19 303 
19 275 
19 247 
19 219 
19 191 



19 164 
19 136 
19 108 
19 080 
19 053 



19 025 
18 997 
18 970 
18 942 
18 914 



18 886 
18 859 
18 831 
18 803 
18 776 



18 748 



Log. Tan. 



Log. Cos, 



92 842 
92 834 
92 826 
92 818 
92 810 



92 802 
92 794 
92 786 
92 778 
92 771 



92 763 
92 755 
92 747 
92 739 
92 731 



92 723 
92 715 
92 707 
92 699 
92 691 



92 683 
92 675 
92 667 
92 659 
92 651 



92 643 
92 635 
92 627 
92 619 
92 611 



92 603 
92 595 
92 587 
92 579 
92 570 



92 562 
92 554 
92 546 
92 538 
92 530 



92 522 
92 514 
92 506 
92 498 
92 489 



92 481 
92 473 
92 465 
92 457 
92 449 



92 441 
92 433 

92 424 
92 416 
92 408 



92 400 
92 392 
92 383 
92 375 
92 367 



9-92 359 
Log. Sin. 



d. 



60 

59 
58 

57 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
II 
30 
29 
28 
27 



25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

_6 

5 
4 
3 
2 
1 





P.P. 





28 


28 


2^ 


6 


2.8 


2.8 


2. 


7 


3 


3 


3 


2 


3. 


8 


3 


8 


3 


7 


3. 


9 


4 


3 


4 


2 


4. 


10 


4 


7 


4 


6 


4. 


20 


9 


5 


9 


3 


9. 


30 


14 


2 


14 





13. 


40 


19 





18 


6 


18. 


50 


23 


7 


23 


3 


22. 





20 


20 


19 


6 


2.0 


2-0 


1.9 


7 


2 


4 


2-3 


2 


3 


8 


2 


7 


2-6 


2 


6 


9 


3 


1 


3-0 


2 


9 


10 


3 


4 


3-3 


3 


2 


20 


6 


8 


6.6 


6 


5 


30 


10 


2 


10.0 


9 


7 


40 


13 


6 


13.3 


13 





50 


17 


1 


16.6 


16 


2 





8 


8 


6 


0-8 


0.8 


7 


1 





0-9 


8 


1 


1 


1.0 


9 


1 


3 


1.2 


10 


1 


4 


1.3 


20 


2 


8 


2.6 


30 


4 


2 


4.0 


40 


5 


6 


5.3 


50 


7 


1 


6.6 



7 

0.7 
0.9 
1.0 
1.1 
1.2 
2.5 
37 
5.0 
6.2 



P.P. 



1»2' 



578 



67"* 



33' 



TABLE VII.— LOGARITHMIC SINES, COSINES, 
AND COTANGENTS. 



TANGENTS. 



146° 



Log. Sin. 



73 
73 
73 
73 
73 



73 
74 
74 
74 
74 

74 
74 
74 
74 
74 



74 
74 
74 
74 
74 



74 
74 
74 
74 
74 



74 
74 
74 
74 
74 



611 
630 
650 
66? 
688 
708 
727 
746 
766 
785 
805 
824 
843 
862 
882 

901 
920 
940 
95? 
978 

997 
Old 
036 
055 
074 

b93 
112 
131 
151 
170 

189 
208 

227 
246 
265 

284 
303 
322 
341 
360 

379 
398 
417 
436 
455 

474 
493 
511 
530 
549 

568 
587 
606 

625 
643 



55 
56 
57 
58 
59 
60 



74 
74 
74 
74 
74 



74 



662 
681 
700 
718 
737 
756 



Log. Cos 



1? 
19 
19 
19 

1? 
19 

1? 
1? 
19 

19 

1? 

19 

1? 
19 

19 
1? 
19 
1? 
19 

19 

1? 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

1? 
19 
19 

19 
18 

19 
19 
19 
19 
19 
19 
19 
18 
19 
19 

1? 
18 
19 
19 
18 

1? 
18 
19 
18 
19 

18 



d. 



Log. Tan. c. d 



251 
279 
307 
334 
362 

390 
417 
445 
473 
500 

528 
555 
583 
610 
638 

666 
693 
721 
748 
776 

803 
831 
858 
886 
913 

941 
968 
996 
023 
051 

078 
105 
133 
160 
188 
215 
243 
270 
297 
325 



352 
380 
407 
434 
462 
48? 
516 
544 
571 
598 

626 
653 
680 
708 
735 

762 
789 
817 
844 
871 



9-82 898 
Log. Cot. 



28 
27 
27 
28 

27 
27 
27 
28 
27 

27 
27 
27 
27 
27 

28 

27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 
27 



c.d, 



Log. Cot. 



18 748 
18 720 
18 693 
18 665 
18 637 



18 610 
18 582 
18 555 
18 527 
18 499 



18 472 
18 444 
18 417 
18 389 
18 362 



18 334 
18 306 
18 27? 
18 251 
18 224 



18 196 
18 16? 
18 141 
18 114 
18 086 



18 05? 
18 031 
18 004 
17 976 
17 949 



17 921 
17 894 
17 867 
17 839 
17 812 



784 
757 
72? 
702 
675 

647 
620 
593 
565 
538 



510 
483 
456 
428 
401 

374 
347 
319 
292 
265 



17 237 
17 210 
17 183 
17 156 
17 128 



17 101 



Log. Tan 



Log. Cos. 



92 359 
92 351 
92 342 
92 334 
92 326 



92 318 
92 310 
92 301 
92 293 
92 285 



92 277 
92 268 
92 260 
92 252 
92 244 



92 235 
92 227 
92 21? 
92 210 
92 202 



92 194 
92 185 
92 177 
92 16? 
92 160 



92 152 
92 144 
92 135 
92 127 
92 119 



92 110 
92 102 
92 094 
92 085 
92 077 



92 06? 
92 060 
92 052 
92 043 
92 035 



92 027 
92 018 
92 010 
92 001 
91 993 



91 984 
91 976 
91 967 
91 959 
91 951 



91 942 
91 934 
91 925 
91 917 
91 908 



91 900 
91 891 
91 883 
91 874 
91 866 



991 857 
Log. Sin. 



60 

59 
58 
57 
-56^ 
55 
54 
53 
52 
IL 
50 
49 
48 
47 

li 

45 
44 
43 
42 
41 

40 

39 
38 
37 

35 
34 
33 

32 
31 

30 

29 
28 
27 
-2i 
25 
24 
23 
22 
21 

20 

19 
18 
17 
11 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6 

5 
4 
3 
2 
1 

O 



P. P. 





28 


37 


21 


6 


28 


2.7 


2. 


7 


3 


2 


3 


2 


3- 


8 


3 


7 


3 


6 


3- 


9 


4 


2 


4 


1 


4. 


10 


4 


6 


4 


6 


4. 


20 


9 


3 


9 


1 


9. 


30 


14 





13 


7 


13. 


40 


18 


6 


18 


3 


18- 


50 


23 


3 


22 


9 


22- 





19 


19 


1^ 


6 


1.9 


1.9 


1. 


7 


2 


3 


2 


2 


2. 


8 


2 


6 


2 


5 


2- 


9 


2 


9 


2 


8 


2. 


10 


3 


2 


3 


1 


3. 


20 


6 


5 


6 


3 


6. 


30 


9 


7 


9 


5 


9. 


40 


13 





12 


6 


12. 


50 


16 


2 


15 


8 


15. 





8 


? 


^ 


6 


0.8 


0.8 


7 


1 








9 


8 


1 


1 


1 





9 


1 


3 


1 


2 


10 


1 


4 


1 


3 


20 


2 


8 


2 


6 


30 


4 


2 


4 





40 


5 


6 


5 


3 


50 


7 


1 


6 


6 



P. p. 



133= 



579 



56^ 



34^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



145° 



Log. Sin. 


9-74 756 


9 


74 775 


9 


74 793 


9 


74 812 


9 


74 831 


9 


74 849 


9 


74 868 


9 


74 887 


9 


74 905 


9 


74 924 


9 


74 943 


9 


74 961 


9 


74 980 


9 


74 998 


9 


75 017 


9 


75 036 


9 


75 054 


9 


75 073 


9 


75 091 


9 


75 110 


9 


75 128 


9 


75 147 


9. 


75 165 


9 


75 184 


9 


75 202 


9. 


75 221 


9 


75 239 


9 


75 257 


9 


75 276 


9 


75 294 


9 


75 313 


9 


75 331 


9 


75 349 


9 


75 368 


9 


75 386 


9 


75 404 


9 


75 423 


9 


75 441 


9 


75 459 


9 


75 478 


9 


75 496 


9 


75 514 


9 


75 532 


9 


75 551 


9 


75 569 


9 


75 587 


9 


75 605 


9 


75 623 


9 


75 642 


9 


75 660 


9 


75 678 


9 


75 696 


9 


75 714 


9 


•75 732 


9 


• 75 750 


9 


•75 769 


9 


•75 787 


9 


•75 805 


9 


•75 823 


9 


•75 841 


9 


75 859 


L 


og. Cos. 



Log. Tan. 



82 898 
82 926 
82 953 

82 980 

83 007 

83 035 
83 062 
83 089 
83 116 
83 143 



c.d. 



83 171 
83 198 
83 225 
83 252 
83 279 



83 307 
83 334 
83 361 
83 388 

83 415 



83 442 
83 469 
83 496 
83 524 
83 551 



83 578 
83 605 
83 632 
83 659 
83 686 



83 713 
83 740 
83 767 
83 794 
83 821 



83 848 
83 875 
83 902 
83 929 
83 9*^7 



83 984 

84 Oil 
84 038 
84 065 
84 091 



84 118 
84 145 
84 172 
84 199 
84 226 

84 253 
84 280 
84 307 
84 334 
84 .^Rl 



84 388 
84 415 
84 442 
84 469 
84 496 



84 R22 



Log. Cot, 



27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
2'7 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
26 

27 
27 
27 
27 
27 

27 
27 
27 
9^ 
26 

27 
27 
27 
27 
27 

26 



Log. Cot. Log. Cos. 



17 101 
17 074 
17 047 
17 019 
16 992 

16 965 



16 938 9 



16 910 
16 883 
16 856 

0^16 829 
0^16 802 
0^16 774 
0.16 74? 
0.16 720 



0^16 693 
0^16 666 
0.16 639 
0^16 612 
0-16 584 



91 857 
91 849 
91 840 
91 832 
91 823 
91 814 
91 806 



9-91 79'7 
9-91 789 
9-91 780 



0.16 557 
0.16 530 
0.16 503 
0.16476 
0^16 449 



91 686 
9^91 677 
91 668 
91 660 
9-91 651 



16 422 
16 395 
16 368 
16 340 
0-16 313 



a. 16 286 
0.16 259 
16 232 
0.16 205 
0.16 178 



0.16 151 
0.16 124 
0.16 097 
0.16 070 
0.16 HAO 



0-16 016 
0.15 989 
0.15 962 
0.15 935 
0.15 90P 



9-91 772 
9-91 763 
9-91 755 
9.91 746 
9-91 737 
9 
9 



91 729 
91 720 
91 712 
91 703 
91 694 



91 642 
9^91 634 
91 625 
91 616 
9-91 608 

9 
9 



91 599 
91 590 
91 582 
91 573 
91 564 



91 556 
9.91 547 
9.91 538 
9.91 529 
9.91 521 



9^91 512 
9 . 91 503 
9 . 91 495 
91 486 
9 . 91 477 



0.15 88T 
0.15 85^ 
0.15 827 
0.15 SO?' 
0.15 773 



15 746 
15 719 
15 699 
15 665 
15 639 



0.15 61? 
0.15 585 
0.15 558 
0-15 531 
0.15 504 

0.15 477 



C.d. I Log, Tan 



91 468 
91 460 

• 91 451 

• 91 442 
91 433 



91 424 
91 416 
9-91 407 
9-91 398 
991 389 



9-91 380 
9-91 372 
9. 91 363 
9-91 354 
9. 91 345 



9. 91 336 



Log. Sin, 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46 



45 
44 
43 
42 
41 

40 

39 
38 
37 

35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 
11 

10 

9 
8 
7 
6 

5 
4 
3 
2 

O 



P. P. 





27 


27 


2 


6 


2.7 


2-7 


2. 


7 


3 


2 


3 


1 


3. 


8 


3 


6 


3 


6 


3. 


9 


4 


1 


4 





4. 


10 


4 


6 


4 


5 


4. 


20 


9 


1 


9 





8. 


30 


13 


7 


13 


5 


13 


40 


18 


3 


18 





17. 


50 


22 


9 


22 


5 


22. 



m 

•71 
81 
91 



10 

§0 
80 
40 
50 



9|0.§ 



21 



}:? 



1.8 
1 
2 

4.2 
5.^ 
7.517.1 





19 


18 


18 ' 


6 


19 


1.8 


18 


7 


2 


2 


2 


1 


2 


1 


8 


2 


5 


2 


4 


2 


4 


9 


2 


8 


2 


8 


2 


7 


10 


3 


1 


3 


1 


3 





20 


6 


3 


6 


1 


6 





30 


9 


5 


9 


2 


9 





40 


12 


6 


12 


3 


12 





50 


15 


8 


15 


4 


15 






p. p. 



IM' 



580 



55' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



144' 



Log. Sin, 



75 859 
75 877 
75 895 
75 913 
75 931 



75 949 
75 967 

75 985 

76 003 
76 021 



76 039 
76 057 
76 075 
76 092 
76 110 



76 128 
76 146 
76 164 
76 182 
76 200 



76 217 
76 235 
76 253 
76 271 
76 289 



76 306 
76 325 
76 342 
76 360 
76 377 



76 395 
76 413 
76 431 
76 448 
76 466 



76 484 
76 501 
76 519 
73 536 
76 554 



76 572 
76 589 
76 607 
76 624 
76 642 



76 660 
76 677 
78 695 
76 712 
76 730 



76 747 
76 765 
76 782 
76 800 
76 817 



76 835 
76 852 
76 869 
76 887 
76 904 



9 76 922 
Log. Cos, 



d. Log. Tan. c.d. Log. Cot. Log. Cos. 



84 5^^ 
84 549 
84 576 
84 603 
84 630 



84 657 
84 684 
84 711 
84 737 
84 7B4 



84 791 
84 818 
84 845 
84 871 
84 898 



84 925 
84 952 

84 979 

85 005 
85 032 



85 059 
85 086 
85 113 
85 139 
85 166 



85 193 
85 220 
85 246 
85 273 
85 300 



85 327 
85 353 
85 380 
85 407 
85 433 



85 460 
85 487 
85 513 
85 540 
85 567 



85 594 
85 620 
85 647 
85 673 
85 700 



85 727 
85 753 
85 780 
85 807 
85 833 



85 860 
85 887 
85 913 
85 940 
85 966 



85 993 

86 020 
86 046 
86 073 
86 099 



86 126 



Log. Cot. 



27 
27 
27 
26 
27 
27 
27 
26 
27 

27 
26 
27 
26 
27 

27 
26 
27 
26 
27 

27 
26 
27 
26 
27 

26 
27 
26 
27 
26 

27 
26 
26 
27 
2'6 

27 
26 
26 
27 
26 

27 
26 
26 
26 
27 

26 
26 
27 
26 
26 

26 
27 
26 
26 
26 

26 
27 
26 
26 
26 
26 



c d, 



15 477 
15 45C 
15 423 
15 396 
15 370 



15 343 
15 316 
15 289 
15 262 
15 235 



15 20C 
15 182 
15 155 
15 128 
15 101 



15 074 
15 048 
15 021 
14 994 
14 967 



14 940 
14 914 
14 887 
14 860 
14 833 



14 807 
14 780 
14 753 
14 726 
14 700 



14 673 
14 646 
14 620 
14 593 
14 566 



14 539 
14 513 
14 486 
14 459 
14 438 



14 406 
14 379 
14 353 
14 326 
14 299 



14 273 
14 246 
14 219 
14 193 
14 166 



14 140 
14 113 
14 086 
14 060 
14 033 



14 007 
13 980 
13 953 
13 927 
13 900 



0-13 874 



Log. Tan, 



91 336 
91 327 
91 318 
91 310 
91 301 



91 292 
91 283 
91 274 
91 265 
91 256 



91 247 
91 239 
91 230 
91 221 
91 212 



91 203 
91 194 
91 185 
91 176 
91 167 



91 158 
91 149 
91 140 
91 131 
91 122 



91 113 
91 104 
91095 
91 086 
91 077 



91 068 
91 059 
91 050 
91 041 
91 032 



91 023 
91 014 
91 005 
90 996 
90 987 



90 978 
90 989 
90 960 
90 951 
90 942 



90 933 
90 923 
90 914 
90 905 
90 896 



90 887 
90 878 
90 869 
90 860 
90 850 



90 841 
90 832 
90 823 
90 814 
90 805 



9.90 798 



x^r 



Log. Sin. 
581 



9 
9 
8 
9 

9 

8 J 
9 
9 
9 

9 
8 
9 
9 
9 
9 
9 
8 
9 
9 
9 
9 
9 
9 
9 

9 
9 
9 
9 
9 

9 
9 
9 
9 
9 
9 
9 
9 
9 
9 

9 
9 
9 
9 
9 
9 
9 
9 
9 
9 

9 



d. 



60 

59 
58 
57 
16 

55 
54 
53 
52 
51 



50 

49 
48 
47 
4S 

45 
44 
43 
42 
41 

40 

39 
38 

37 
36 

35 
34 
33 
32 
11 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
11 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6 



P. P. 



27 



2 


7 


2- 


3 


1 


3- 


3 


6 


3- 


4 





4- 


4 


5 


4. 


9 





8. 


13 


5 


13- 


18 





17. 


22 


5 


22. 



26_ 

6 
1 
5 

4 
8 
2 
6 
1 





18 


17 




17 


f 


6 


1.8 


1.7 


1.7 


7 


2 


1 


2 





2 





8 


2 


4 


2 


3 


2 


2 


9 


2 


7 


2 


6 


2 


5 


]0 


3 





2 


9 


2 


8 


20 


6 





5 


8 


5 


6 


30 


9 





8 


7 


8 


5 


40 


12 


C 


11 


6 


11 


3 


50 


15 





14 





14 


1 





9 




9 




6 


0.0 


0.9 





7 


1.1 


1 





1 


8 


1.2 


1 


2 


1- 


9 


1.4 


1 


3 


1 


10 


i.e 


1 


5 


1. 


20 


3.1 


3 





2. 


30 


4.7 


4 


5 


4. 


40 


3.3 


6 





5. 


50 


7.9 


7 


5 


7. 



p.p. 



5i' 



36' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



143'* 



Log. Sin, 



76 922 
76 939 
76 956 
76 974 
76 991 



77 008 
77 026 
77 043 
77 060 
77 078 



10 

11 
12 
13 
li 
15 
16 
17 
18 
19_ 

30 

21 
22 
23 
24 9 



25 

26 

27 

28 

29_ 

30 

31 

32 

33 

31 

35 

36 

37 

38 

39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 
51 
52 
53 

54 

55 
58 
57 
58 
5Q 



77 095 
77 112 
77 130 
77 147 
77 164 



77 181 
77 198 
77 216 
77 233 
77 250 



77 267 
77 284 
77 302 
77 319 
77 336 



77 353 
77 370 
77 387 
77 404 
77 421 



77 439 
77 456 
77 473 
77 490 
77 507 



77 524 
77 541 
77 558 
77 575 
77 592 



77 609 
77 626 
77 643 
77 660 
77 677 



77 693 
77 710 
77 727 
77 744 
77 781 



77 778 
77 795 
77 812 
77 828 
77 845 



77 8S2 
77 879 
77 896 
77 91R 
77 929 



9-77 9A6 



17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 
17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 
17 
17 
17 
17 
17 

17 
17 
17 
17 
17 
16 
17 
17 
17 
17 

16 
17 
17 
16 
17 

17 
16 
17 
17 
16 

17 



Log. Cos 



Log. Tan. 



86 126 
86 152 
86 179 
86 206 
88 232 



9-86 259 
9-86 285 
9-86 312 
9-86 338 
9.86 365 



9-86 391 
9-86 418 
9.86 444 
9.86 471 
9.86 497 



9-86 524 
86 550 
9.86 577 
9-86 603 
86 630 



9.86 656 
9.86 683 
9.86 709 
9.86 736 
9-86 762 



c. d. 



9.86 788 
9.86 815 
9.86 841 
9. 86 868 
9. 86 894 



9-86 921 

9.86 947 

86 973 

87 000 
87 026 

9.87 053 
9.87 079 
9-87 105 
9.87 132 
9.87 158 



9-87 185 
9.87 211 
9.87 237 
9.87 264 
9-87 290 



87 316 
87 343 
87 369 
87 395 
87 422 



9.87 448 
9.87 474 
9.87 501 
9-87 527 
9-87 553 
9 
9 



87 580 
87 eOfi 
87 632 
87 659 
87 685 



9. 87 71] 



d. Log. Cot. 



26 
26 
27 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 

26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
28 
26 
26 

26 
cTd 



0.13 874 
0.13 847 
13 821 
13 794 
0.13 767 



Log. Cot. 



9-90 796 
90 786 
9-90 777 
9. 90 768 
9.90 759 



13 741 
13 714 
13 688 
13 66l 
13 635 



0.13 608 
0.13 582 
0.13 555 
0.13 529 
0.13 502 



Log. Cos. 



90 750 
90 740 
90 73l 
90 722 
90 713 



9.90 703 
9 . 90 694 
9. 90 685 
90 676 
9. 90 666 



0.13 476 
0.13 449 
0-13 423 
0.13 396 
013 370 



90 657 
90 648 
9-90 639 
9-90 629 
9.90 620 



13 343 
13 317 
13 290 
13 264 
13 237 



13 211 
13 185 
13 158 
13 132 
13 10 



13 079 
13 052 
13 026 
13 000 
12 973 

12 947 
12 920 
12 894 
12 868 
12 84l 



9-90 611 
9-90 602 
9.90 592 
9.90 583 
9.90 574 



9-90 564 
9-90 555 

90 546 
9-90 536 

90 527 



12 815 
12 78S 
0.12 762 
0-12 73e 
0-12 7C8 



12 683 
12 657 
12 63C 
12 60? 
12 57£ 



12 551 
12 52f 
12 499 
12 472 
12 446 



12 420 
12 393 
12 367 
12 341 
12 315 

12 28§ 



Log. Tan 



9-90 518 
9-90 508 
9-90 499 
90 490 
9 - 90 480 

9-90 471 
90 461 
90 452 
90 443 
80 433 



90 424 
9-90 414 
9- 90 405 
9. 90 396 
9-90 386 



90 377 
90 367 
90 358 
90 348 
90 339 



90 330 
9-90 320 
9-90 311 
9-90 301 
9.90 292 



9-90 282 
9. 90 273 
9. 90 263 
9-90 254 
9-90 244 



9-90 235 



X36" 



Log. S 
582 



in. 



d. 



60 

59 
58 
57 
56 



50 

49 
48 

47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36_ 
35 
34 
33 
32 
31 



SO 

29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 
11 

10 

9 
8 

7 
6 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



17 

l-"7 



14.6 



17 

1 

2 

2 

2 

2 

5 

8 
11 
14 



16_ 

1.6 
1.9 
2.2 
2.5 
2.7 
55 
8.2 
311.0 
113.7 



6 

7 

8 

9 

IC 

20 

30 

40 

50 



fl 


) 


0.91 


1 


1 


1 


2 


1 


4 


1 


6 


3 


1 


4 


7 


6 


3 


7 


9 



9 

0.9 
1.0 
1.2 
1-3 
1.5 
3.0 
4.5 
6.0 
7.5 



P. P, 



27 


36 


26 


2-7 


2.6 


2.6 


3 


1 


3.1 


3.0 


3 


6 


3.5 


3.4 


4 





4.0 


3.9 


4 


5 


4.4 


4.3 


9 





8.8 


86 


13 


5 


13.2 


13-0 


18 


■ 


17-6 


17-3 


22 


• 5 


22.1 


21.6 



63' 



37' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 143° 



Log. Sin. 



77 946 
77 963 
77 980 

77 996 

78 013 



78 030 
78 046 
78 063 
78 080 
78 097 



78 113 
78 130 
78 147 
78 163 
78 180 



78 196 
78 213 
^8 230 
78 246 
78 263 



78 279 
78 296 
78 312 
78 329 
78 346 



78 362 
78 379 
78 395 
78 412 
78 428 



78 444 
78 461 
78 477 
78 494 
78 510 



78 527 
78 543 
78 559 
78 576 
78 592 



78 609 
78 625 
78 64l 
78 658 
78 674 



78 690 
78 707 
78 723 
78 739 
78 755 



78 772 
78 788 
78 804 
78 821 
78 837 



78 853 
78 869 
78 885 
78 902 
78 918 

78 934 



Log. Cos. 



16 
17 
16 
17 
16 
16 

17 
16 
17 

16 
16 
17 
16 
1'6 
16 
16 
17 
16 
16 

16 
16 
16 
16 
17 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 

16 
16 
16 

16 
16 
16 
16 
16 
16 



Log. Tan, 



87 711 
87 737 
87 764 
87 790 
87 816 



87 843 
87 869 
87 895 
87 921 
87 948 



87 974 

88 000 
88 026 
88 053 
88 079 



88 105 
88 131 
88 157 
88 184 
88 210 



88 236 
88 262 
88 288 
88 315 
88 341 



88 367 
88 393 
88 419 
88 445 
88 472 



88 498 
88 524 
88 550 
88 576 
88 602 



88 629 
88 655 
88 681 
88 707 
88 733 



88 759 
88 785 
88 811 
88 838 
88 864 



88 890 
88 916 
88 942 
88 968 
88 994 



89 020 
89 046 
89 072 



c.d.jLog, Cot, 



89 150 
89 177 
89 203 
89 229 
89 255 



26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 

26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 



89 098i o^ 
89 124 ^^ 



89 281 



26 
26 
26 
26 
26 

26 



Log. Cot. c.d. 



|0.12 288 
|0.12 262 
10.12 236 
|0.12 209 
;0.12 183 



12 157 
12 131 
12 104 
12 078 
12 052 



0.12 026 
0.11 999 
0.11 973 
0-11 947 
0.11 921 



0.11 895 
0.11 868 
0.11 842 
0.11 816 
Oil 790 



11 763 
11 737 
11 711 
11 685 
11 659 



iO-ll 
0.11 
0.11 
iO.ll 
0.11 



lO.ll 
0-11 
0.11 
0.11 
0-11 



633 
606 
580 
554 
528 
502 
476 
449 
423 
397 9 



Log. Cos. 



0.11 371 
0.11 345 
0.11 319 
0.11 293 
0.11 266 



0.11 240 
0.11 214 
0.11 188 
0.11 162 
0.11 136 



11 110 
11 084 
11 058 
11 032 
11 005 



10 979 
10 953 
0-10 927 
0.10 901 
0.10 875 



10 849 
10 823 
010 797 
010 771 
10 745 



0.10 719 



Log. Tan, 



90 235 
90 225 
90 216 
90 206 
90 196 



90 187 
90 177 
90 168 
90 158 
90 149 



90 139 
90 130 
90 120 
90 110 
90 101 



90 091 
90 082 
90 072 
90 062 
90 053 



90 043 
90 033 
90 024 
90 014 
90 004 



89 995 
89 985 
89 975 
89 966 
89 956 



89 946 
89 937 
89 927 
89 917 
89 908 



89 898 
89 888 
89 878 
89 869 
89 859 



89 849 
89 839 
89 830 
89 820 
89 810 



89 800 
89 791 
89 781 
89 771 
89 761 



89 751 
89 742 
89 732 
89 722 
89 712 



89 702 
89 692 
89 683 
89 673 
89 663 



89 653 



•og. Sin. 



9 

9 

9 

10 

9 
9 
9 
9 
9 

9 
9 
10 
9 
9 

9 
9 
10 
9 
9 

9 

10 

9 

9 

10 

9 
9 
10 
9 
9 

10 
9 
9 

10 
9 

10 
9 

10 
9 

10 

9 
10 

9 
10 

9 

10 

9 

10 

10 

9 

10 

9 

10 

10 

9 

10 
10 
9 
10 
10 
10 



60 

59 
58 
57 
-56_ 
55 
54 
53 
52 
-51 
50 
49 
48 
47 

45 
44 
43 
42 
41 



d. 



40 

39 
38 

37 
_36 

35 
34 
33 
32 
-31 
30 
29 
28 
27 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 
11 

10 

9 

8 

7 

_i 

5 
4 
3 
2 
_X 
O 



P.P. 





26 


2 


6 


2.6 


2 


7 


3 


1 


3. 


8 


3 


5 


3- 


9 


4 





3. 


10 


4 


4 


4. 


20 


8 


8 


8. 


30 


13 


2 


13. 


40 


17 


6 


17. 


50 


22 


1 


21. 





17 


16 


1( 


6 


17 


1.6 


1. 


7 


2 





1-9 


1- 


8 


2 


2 


2.2 


2. 


9 


2 


5 


2.5 


2. 


10 


2 


8 


2.7 


2. 


20 


5 


6 


5-5 


5. 


30 


8 


5 


8.2 


8. 


40 


11 


3 


11.0 


10. 


50 


14 


1 


13.7 


13. 



10 9 



09 
1.1 
1.2 



61.0 

7|l.I 

81.3 

91.5 1.4 
101.611.6 
203-3 3.1 
305.0 4.7 
406.6 6.3 
508.3 7-9 



P. P. 



137° 



583 



52' 



38* 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. ' i41« 





1 

2 

3 

_4 

5 
6 
7 
8 
Ji 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

30 

21 
22 
23 

24 

25 
26 
27 
28 
2i 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 

47 

48 

4£L 

60 

51 

52 

53 

51 

55 
56 
57 
58 
59. 
60 



Log. Sin. 



78 934 
78 950 
78 966 
78 982 
78 999 



79 015 
79 031 
79 047 
79 083 
79 079 



79 095 
79 111 
79 127 
79 143 
79 159 



79 175 
79 191 
79 207 
79 223 
79 239 



79 255 
79 271 
79 287 
79 303 
79 319 

79 335 
79 351 
79 367 
79 383 
79 399 



79 415 
79 431 
79 446 
79 462 
79 478 



79 494 
79 510 
79 526 
79 541 
79 557 



79 573 
79 589 
79 605 
79 620 
79 636 



79 652 
79 668 
79 683 
79 699 
79 715 



79 730 
79 746 
79 762 
79 777 
79 793 



9. 79 887 



79 809 
79 824 
79 840 
79 856 
79 871 



Log. Cos. 



16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
15 
16 

16 
16 
15 
16 
16 

15 
16 
16 
15 
16 

16 
15 
16 
15 
16 

15 
16 
15 
16 
15 

15 
16 
15 
15 
16 

15 
15 
16 
15 
15 

15 



Log. Tan 



89 281 
89 307 
89 333 
89 359 
89 385 



89 411 
89 437 
89 463 
89 489 
89 515 



89 541 
89 567 
89 593 
89 619 
89 645 



89 671 
89 697 
89 723 
89 749 
89 775 
89 801 
89 827 
89 853 
89 879 
89 905 



89 931 
89 957 

89 982 

90 008 
90 034 



90 060 
90 086 
90 112 
90 138 
90 164 



90 190 
90 216 
90 242 
90 268 
90 294 



90 319 
90 345 
90 371 
90 397 
90 423 



90 449 
90 475 
90 501 
90 526 
90 552 



90 578 
90 604 
90 630 
90 658 
90 682 



90 707 
90 733 
90 759 
90 785 
90 811 



90 837 



d. Log. Cot. c.d 



c.d, 



26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
25 
26 
26 
26 
26 
26 
26 
25 

26 
26 
26 
26 
26 
25 
26 
26 
26 
25 

26 
26 
26 
25 
26 

26 
26 

25 
26 
26 

25 
26 
26 
26 
25 
26 



Log. Cot, 



10 719 
10 693 
10 667 
10 641 
10 615 



10 589 
10 563 
10 537 
10 511 
10 485 



10 459 
10 433 
10 407 
10 381 
10 3^5 

10 329 
10 303 

10 277 
10 251 
10 225 



10 199 
10 173 
10 147 
10 121 
10 095 



Log. Cos. 



10 089 
10 043 
10 017 
09 991 
09 965 



09 939 
09 913 
09 887 
09 861 
09 838 



09 810 
09 784 
09 758 
09 732 
09 706 



09 680 
09 654 
09 628 
09 602 
09 577 



09 551 
09 525 
09 499 
09 473 
09 447 



09 421 
09 395 
09 370 
09 344 
09 318 



009 163 



09 292 
09 286 
09 240 
09 214 
09 189 



Log. Tan. 



89 653 
89 643 
89 633 
89 623 
89 613 



89 604 
89 594 
89 584 
89 574 
89 564 



89 554 
89 544 
89 534 
89 524 
89 514 



89 504 
69 494 
89 484 
89 474 
89 464 



89 454 
89 444 
89 434 
89 424 
89 414 



89 404 
89 394 
89 384 
89 374 
89 364 



89 354 
89 344 
89 334 
89 324 
89 314 

89 304 
89 294 
89 284 
89 274 
89 264 



89 253 
89 243 
89 233 
89 223 
89 213 



89 203 
89 193 
89 182 
89 172 
89 162 



89 152 
89 142 
89 132 
39 121 
89 111 



89 101 
89 091 
89 081 
89 070 
89 060 



9-89 050 



Log. Si 



9 
10 
10 
10 

9 
10 
10 
10 
10 

10 
9 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 

10 
10 
10 

10 
10 
10 
10 
10 

10 



d. 



60 

59 
58 
57 
56_ 

55 
54 
53 
52 
II 
50 
49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
16 

35 
34 
33 
32 
II 
30 
29 
28 
27 
21 
25 
24 
23 
22 
21. 
30 
19 
18 
17 
16, 
15 
14 
13 
12 

10 

9 
8 
7 
_6. 

5 
4 
3 
2 
l_ 




P.P. 





26 


6 


2.61 


7 


3 





8 


3 


4 


9 


3 


9 


10 


4 


3 


20 


8 


6 


30 


13 





40 


17 


3 


50 


21 


6 



35 

2.5 
30 



3 

3 

4 

8 
12 
17.0 
21.2 



6 

7 

8 

9 

10 

20 

30 

40 

50 



lO 10 9 








1-0 







2 


1-1 


1- 




4 


1-3 


1- 




6 


15 


1- 




7 


1-6 


1. 


3 


5 


3-3 


3- 


5 


2 


5.0 


4. 


7 





6-6 


6. 


8 


7 


8.3 


7. 















1 


16 16 15 


6 


1.6 


1-6 


1-5 


7 


1 


9 


1 


8 


1 


8 


8 


2 


2 


2 


1 


2 





9 


2 


5 


2 


4 


2 


3 


10 


2 


7 


2 


6 


2 


6 


20 


5 


5 


5 


3 


5 


I 


30 


8 


2 


8 





7 


7 


40 


11 


10 


6 


10 


3 


50 


13 


7 


13 


3 


12 


9 



l'^ 



p. p. 



138= 



584 



51' 



TABLE VII.— LOGARITHMIC SINES, COSINES, 
AND COTANGENTS. 



TANGENTS. 



140' 



Log. Sin, 



79 887 
79 903 
79 918 
79 934 
79 949 



79 965 
79 980 

79 996 

80 011 
80 027 



80 042 
80 058 
80 073 
80 089 
80 104 



80 120 
80 135 
80 151 
80 166 
80 182 



80 197 
80 213 
80 228 
80 243 
80 259 



80 274 
80 289 
80 305 
80 320 
80 335 



80 351 
80 366 
80 381 
80 397 
80 412 



80 427 
80 443 
80 458 
80 473 
80 488 



80 504 
80 519 
80 534 
80 549 
80 554 



80 580 
80 595 
80 610 
80 625 
80 640 



80 655 
80 671 
80 686 
80 701 
80 716 



80 731 
80 746 
80 761 
80 776 
80 791 



9. 80 806 



Log. Cos 



Log. Tan. c. d.Log. Cot, 



90 837 
90 863 
90 888 
90 914 
90 940 



90 966 

90 992 

91 017 
91 043 
91 069 



91 095 
91 121 
91 146 
91 172 
91 198 



91 224 
91 250 
91 275 
91 301 
91 327 



91 353 
91 378 
91 404 
91 430 
91 456 



91481 
91 507 
91 533 
91 559 
91 584 



91 610 
91 636 
91 662 
91 687 
91 713 



91 739 
91 765 
91 790 
91 816 
91 842 



91 867 
91 893 
91 919 
91 945 
91 970 



91 996 

92 022 
92 047 
92 073 
92 099 



92 124 
92 150 
92 176 
92 201 
92 227 



92 253 
92 278 
92 304 
92 330 
92 355 



92 381 
Log. Cot. 



26 
25 
26 
25 

26 
26 
25 
26 
26 
25 
26 
25 
26 
25 

26 
26 
25 
26 
25 

26 
25 
26 
25 
26 

25 
26 
25 
26 
25 

26 
25 
26 
25 
26 

25 
26 
25 
26 
25 

25 
26 
25 
26 
25 

25 
26 
25 
26 
25 

25 
26 
25 
25 
26 

25 
25 
26 
25 
25 

26 
c. d. 



09 163 
09 137 
09 111 
09 085 
09 060 



09 034 
09 008 
08 982 
08 956 
08 930 



08 905 
08 879 
08 853 
08 827 
08 802 



08 776 
08 750 
08 724 
08 698 
08 673 



08 647 
08 621 
08 595 
08 570 
08 544 



08 518 
08 492 
08 467 
08 441 
08 415 



08 389 
08 364 
08 338 
08 312 
08 286 



08 261 
08 235 
08 209 
08 183 
08 15£ 



08 132 
08 106 
08 081 
08 055 

08 029 



08 004 
07 978 
07 952 
07 926 
07 901 



07 875 
07 849 
07 824 
07 798 
07 772 



07 747 
07 721 
07 695 
07 670 
07 644 



0-07 618 
Log. Tan. 



Log. Cos, 



89 050 
89 040 
89 030 
89 019 
89 009 
88 999 
88 989 
88 978 
88 968 
88 958 



88 947 
88 937 
88 927 
88 917 
88 906 



88 896 
88 886 
88 875 
88 865 
88 855 



88 844 
88 834 
88 823 
88 813 
88 803 



88 792 
88 782 
88 772 
88 76l 
88 751 



88 740 
88 730 
88 720 
88 709 
88 699 

88 688 
88 678 
88 667 
88 657 
88 646 



88 636 
88 625 
88 615 
88 604 
88 594 



88 583 
88 573 
88 562 
88 552 
88 54l 



88 531 
88 520 
88 510 
88 499 
88 489 



88 478 
88 467 
88 457 
88 446 
88 436 



9 88 425 
Log. Sin. 



d. 



10 
10 
lO 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
•10 
10 
10 
10 

10 
10 
10 
10 
10 

10 
10 
10 
10 
10 

10 

11 

10 
10 
10 

10 



60 

59 
58 
57 
16 
55 
54 
53 
52 
_51 
50 
49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 

li 

35 
34 
33 
32 
11 
30 
29 
28 
27 
_26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
-16 

15 
14 
13 
12 
U 

10 

9 
8 

7 
_6 

5 
4 
3 

2 





P. p. 



26 



2 


6 


2. 


3 





3. 


3 


4 


3. 


3 


9 


3- 


4 


3 


4. 


8 


6 


8- 


13 





12. 


17 


3 


17. 


21 


6 


21. 



25 

5 

4 
8 
2 
5 
7 

2 





16 


15 


1 


6 


16 


1.5 


1. 


7 


1 


8 


1 


8 


1. 


8 


2 


1 


2 





2. 


9 


2 


4 


2 


3 


2. 


10 


2 


6 


2 


6 


2. 


20 


5 


3 


5 


1 


5. 


30 


8 





7 


7 


7- 


40 


10 


6 


10 


3 


10. 


50 


13 


3 


12 


9 


12. 





11 


10 




6 


1-1 


1-0 




7 




3 




2 




8 




4 




4 




9 




6 




6 




10 




8 




7 




20 


3 


6 


3 


5 


3. 


30 


5 


5 


5 


2 


5. 


40 


7 


3 


7 





6. 


50 


9 


1 


8 


7 


8. 



P. p. 



139° 



585 



60** 



40' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



139' 



15 9 
9 
9 
9 



Log. Sin, 



80 806 
80 822 
80 837 
80 852 
80 867 



80 882 
80 897 
80 912 
80 927 
80 942 



80 957 
80 972 

80 987 

81 001 
81 016 



81 031 
81 046 
81 061 
81 076 
81 091 



81 106 
81 121 
81 138 
81 150 
81 165 



81 180 
81 195 
81 210 
81 225 
81 239 



81 254 
81 289 
81 284 
81 299 
81 313 



81 328 
81 343 
81 358 
81 372 
81 387 



81 402 
81 416 
81 431 
81 448 
81 4fi0 



81 475 
81490 
81 504 
81 519 
81 534 



81 548 
81 563 
81 578 
81 592 
81 607 



81 621 
81 633 
81 650 
81 665 
81 680 
9-81 694 



Log. Cos, 



15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
14 
15 

15 
15 
15 
15 
14 

15 
15 
15 
14 
15 

15 
14 
15 
15 
14 

15 
14 
15 
15 
14 

15 
14 
15 
14 
14 

15 
14 
15 
14 
14 

15 
14 
14 
15 
14 

14 
14 
15 
14 
14 

14 
14 
14 
14 
15 
14 



d. 



Log. Tan. c. d. 



92 381 
92 407 
92 432 
92 458 
92 484 



92 509 
92 535 
92 561 
92 586 
92 612 



92 638 
92 683 
92 689 
92 714 
92 740 



92 783 
92 791 
92 817 
92 842 
92 S^H 



92 894 
92 9191 
92 945 
92 971 
92 996 



93 022 
93 047 
93 073 
93 098 
93 124 



93 150 
93 175 
93 201 
93 226 
93 252 



93 278 
93 303 
93 329 
93 354 
93 380 



93 405 
93 431 
93 456 
93 482 
93 508 



93 533 
93 559 
93 584 
93 610 
93 635 



93 661 
93 686 
93 712 
93 737 
93 763 



93 788 
93 814 
93 840 
93 865 
93 891 



93 916 



25 
25 
26 
25 

25 
25 
28 
25 
25 

26 

25 
25 
25 
26 

25 
25 
25 
25 
23 

25 
25 
25 
26 
25 

25 
25 
25 
25 
26 

25 
25 
25 
25 
25 

26 

25 
25 
25 
25 

25 
25 
25 
25 
26 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
26 
25 

25 

25 



Log. Cot. 



07 362 
07 336 
07 311 
0-07 285 
0.07 259 



07 618 
07 593 
07 587 
07 541 
07 516 



07 490 
07 4G5 
07 439 
07 413 
07 388 9 



Log. Cos. d 



0-07 234 
0.07 208 
007 183 
0-07 157 
007 131 



07 106 
0.07 080 
007 055 
0-07 029 
007 003 



006 978 
006 952 
0-06 927 
0-08 901 
0-06 875 



006 850 
008 824 
0-08 799 
008 773 
008 748 



06 722 
08 896 
0-08 871 
0-08 845 
006 620 



008 594 
0*06 569 
0.08 543 
008 518 
008 492 



006 486 
. 08 441 
0-08 415 
006 390 
0-06 38A 



08 339 
06 313 
0-08 288 
0-08 282 
006 237 



0-06 211 
006 186 
0-06 160 
0-08 134 
0-08 109 



0-08 083 



88 425 
88 415 
88 404 
88 393 
88 383 



88 372 
88 361 
88 351 
88 340 
88 329 



88 319 
88 308 
88 297 
88 287 
88 276 



88 265 
88 255 
88 244 
88 233 
88223 



88 212 
88 201 
88 190 
88 180 
88 169 



88 158 
88 147 
88 137 
88 126 
88 115 



88 104 
88 094 
88 083 
88 072 
88 061 



88 050 
88 039 
88 029 
88 018 
88 007 



87 996 
87 985 
87 974 
87 963 
87 953 



87 942 
87 931 
87 920 
87 909 
87 898 



87 887 
87 876 
87 865 
87 854 
87 844 



87 833 
87 822 
87 811 
87 800 
87 789 



87 778 



130' 



Log. Cot. c.d.|Log. Tan. [Log. Sin. 
586 



10 

11 

10 
10 

10 

11 

10 
10 

11 

10 
10 

11 

10 
10 

11 

10 
10 

11 

10 

11 

10 

11 

10 

11 

10 

11 

10 

11 

10 

11 

10 

11 
11 

10 

11 
11 

10 

11 
11 

10 

11 
11 
11 

10 

11 
11 
11 

10 

11 

11 
11 
11 
11 

10 

11 
11 
11 
11 
11 

11 



60 

59 

58 

57 

_56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
48 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
IL 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 
11 

10 

9 
8 

7 
_6^ 

5 

4 
3 
2 

O 



P. P. 





36 


6 


2.81 


7 


3 





8 


3 


4 


9 


3 


9 


10 


4 


3 


20 


8 


6 


30 


13 





40 


17 


3 


50 


21 


6 



25 

2-5 





15 


15 


1 


f 


6 


1.5 


1.5 


1 


4 


7 


1 


8 


1 


7 


1 


7 


8 


2 





2 


c 


1 


9 


9 


2 


3 


2 


2 


2 


2 


10 


2 


8 


2 


5 


2 


4 


20 


5 


1 


5 





4 


8 


30 


7 


7 


7 


5 


7 


2 


40 


10 


3 


10 





9 


6 


50 


12 


9 


12 


5 


12 


1 



11 



6 


1 


1 


1.0 


7 




3 


1.2 


8 




4 


1.4 


9 




6 


1-6 


10 




8 


1.7 


20 


3 


6 


3.5 


30 


5 


5 


5.2 


40 


7 


3 


70 


50 


9 


1 


8.7 



10 



p. p. 



49^ 



41^ 



TABLE Vn.— LOGARITHMIC SINES, COSINES, TANGENTS. 

AND COTANGENTS. 13go 



O 

1 

2 

3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 
Ik 
15 
16 
17 
18 

11 
20 

2J 



Log. Sin. 



81 694 
81 709 
81 723 
81 738 
81 752 



81 767 
81 781 
81 796 
81 81 
81 824 



81 839 
81 853 
81 868 
81 882 
81 897 



23 9 

24 9 

25 
26 
27 
28 
29 



30 

31 

32 
33 
34 

35 
38 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59_ 
60 



81 911 
81 925 
81 940 
P-1 954 
81 969 

81 983 

81 997 

82 012 
82 026 
82 040 



9-82 055 
9. 82 069 
9-82 083 
9. 82 098 
9-82 112 



82 126 
82 140 
82 155 
82 169 
82 183 



82 197 
82 212 
82 226 
82 240 
82 254 



82 209 
82 283 
82 297 
82 311 
82 325 



82 339 
82 354 
82 368 
82 382 
82 396 



82 410 
82 424 
82 438 
82 452 
82 467 



82 481 
82 495 
82 509 
82 523 
82 537 



9. 82 551 



Log. Cos, 



14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 



Log. Tan. 



9-93 916 
93 942 
93 967 
9-93 993 
9^^94 018 

9 . 94 044 
9 94 069 
9. 94 095 
9. 94 120 
9. 94 146 



c.d, 



94 171 
94 197 
94 222 
94 248 
94 273 



94 299 
94 324 
9-94 350 
9-94 375 
9 . 94 400 



■94 426 

•94 451 

94 477 

• 94 502 

• 94 528 



94 553 
9^94 579 
9 . 94 804 
9-94 630 
9-94 655 



9-94 681 
9-94 706 
9-94 732 
9- 94 757 
9-94 782 



94 808 
94 833 
94 859 
94 884 
94 910 



d. 



94 935 
94 961 
94 986 
9-95 Oil 
995 037 



95 062 
95 088 
9-95 113 
95 139 
95 164 



95 189 
95 215 
95 240 
95 266 
95 291 



95 316 
9- 95 342 
995 367 

95 393 
9-95 418 



9-95 443 



Log. Cot. 



25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 



Log. Cot 



0-06 083 
0- 06 058 
0-06 032 
0- 06 007 
005 981 



0-05 956 
0-05 930 
0-05 905 
005 879 
0-05 854 



0-05 82e 
0-05 8GS 
0-05 777 
0-05 752 
0-C5 726 



0-05 701 
0-05 675 
05 650 
0-05 62f 
0_05 5PQ 



0-05 574 
05 548 
0-05 523 
0-05 497 
0-05 472 



0-05 446 
005 42] 
0-05 395 
05 37C 
0-05 34^ 



0-05 319 
0-05 293 
05 268 
005 243 
0-05 217 



C-05 192 
005 166 
0-05 141 
0-05 115 
0.05 090 



Log. Cos. 



87 778 
87 767 
87 756 
87 745 
87 734 



87 723 
87 712 
87 701 
87 690 
87 679 



87 668 
87 657 
87 645 
87 634 
87 623 



9-87 612 
9-87 601 
9-87 590 
9-87 579 
9- 87 568 



9-87 557 
9-87 546 
9-87 535 
9-87 523 
987 512 



9-87 501 
9-87 290 
9. 87 479 
9. 87 468 
9-87 457 



87 445 
87 434 
87 423 
9-87 412 
9-87 401 



c.d, 



0-05 064 
0-05 039 
0- 05 014 
0. 04 988 
0-04 963 



0-04 937 
0-04 912 
0-04 886 
0-04 861 
0-04 836 



0-04810 
0-04 785 
0-04 759 
0-04 734 
0-04 708 



04 683 
04 658 
004 632 
0-04 607 
0-04 581 



0-04 556 



87 389 
87 378 
9-87 367 
87 356 
87 345 



9-87 333 
9-87 322 
9-87 311 
9-87 300 
9-87 288 



87 277 
87 266 

9-87 254 
87 243 

987 232 



87 221 
87 209 
87 198 
87 187 
87 175 



9-87 164 
9-87 153 
9-87 141 
87 130 
87 118 



d. 



9-87 107 



Log. Tan. [Log. Sin, 



11 
11 
11 
11 
11 
11 
11 
11 
11 

11 
11 
ll 
11 
11 

11 
11 
11 
11 
11 

11 
11 
11 
11 
11 

11 
11 
11 
11 
11 

ll 
11 
11 
11 
11 

ll 
11 
11 
11 
11 

ll 
11 
ll 
11 
11 

ll 
11 
11 
11 
11 

11 
11 
ll 
11 
11 

ll 
11 
11 
11 
11 
11 



60 

59 
58 
57 

55 
54 
53 
52 

50 

49 
48 

47 
-46 

45 
44 
43 
42 
41 

40 

39 
38 

37 
86 

35 
34 
33 
32 
31 

30 

29 
28 
27 
26_ 

25 
24 
23 
22 

20 

19 
18 

17 
-16 
15 
14 
13 
12 
11 



10 

9 
8 
7 
_6 

5 
4 
3 

2 
1 



P.P. 





25 


25 


6 


2 


5 


2-5 


7 


3 





2 


9 


8 


3 


4 


3 


3 


9 


3 


8 


3 




10 


4 


2 


4 


] 


20 


8 


5 


8 


3 


30 


12 


7 


12 




40 


17 





16 


^ 


50 


21 


2 


20 


8 





15 


14 


6 


1 -4 


1-4 


7 


1 


7 


1 


f5 


8 


1 


9 


1 


8 


9 


2 


2 


2 


1 


10 


2 


4 


2 


— 


on 


4 




4 


P 


30 


7 


2 


7 





40 


9 


6 


9 


^ 


50 


12 


1 


11 


6 





11 


11 


6 


1-1 


1-1 


7 




3 


1 


3 


8 




5 


1 


4 


9 




7 


1 


6 


10 


-] 


9 


1 


8 


20 


3 


8 


3 


g 


30 


5 


7 


5 


5 


40 


7 


6 


7 


3 


50 


9 


6 


9 


1 



P.P. 



131' 



587 



48 



42' 



TABLE VII.— LOGARITHMIC SINES, COSINES, 
AND COTANGENTS. 



TANGENTS 



137° 



O 

1 

2 

3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 



Log. Sin. 



4£9 

45 
46 
47 
48 
49^ 

50 

51 

52 

53 

5A 

55 

56 

57 

58 

59_ 

60 



551 
565 
579 
593 
607 



621 
635 
649 
663 
677 
691 
705 
719 
733 
746 



760 
774 
788 
802 
816 



830 
844 
858 
871 
885 



899 
913 
927 
940 
954 



968 
982 
996 
009 
023 



037 
051 
064 
078 
092 



106 
119 
133 
147 
160 



174 
188 
20l 
215 
229 



242 
256 
269 
283 
297 



310 
324 
337 
351 
365 



83 378 



Log. Cos. 



14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
13 

14 
14 
14 
14 
13 

14 
14 
14 
13 
14 

14 
13 
14 
13 
14 

14 
13 
14 
13 
14 

13 
14 
13 
14 
13 

14 
13 
13 
14 
13 

13 
14 
13 
13 
14 

13 
13 
13 
14 
13 

13 
13 
13 
13 
14 

13 



Log. Tan. 



95 443 
95 469 
95 494 
95 520 
95 545 



95 571 
95 596 
95 62l 
95 647 
95 672 



95 697 
95 723 
95 748 
95 774 
95 799 



95 824 
95 850 
95 875 
95 901 
95 926 



95 951 

95 977 

96 002 
96 027 
96 053 



96 078 
96 104 
96 129 
96 154 
96 180 



96 205 
96 230 
96 256 
96 281 
96 306 



96 332 
96 357 
96 383 
96 408 
96 433 



96 459 
96 484 
96 509 
96 535 
96 560 



96 585 
96 611 
96 636 
96 661 
96 687 



96 712 
96 737 
96 763 
96 788 
9111_3 
96 839 
96 864 
96 889 
96 915 
96 940 



9-96 965 



C. d. 

25 
25 
25 
25 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 

25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 



Log. Cot, 



Log. Cot. 



556 
531 
505 
480 
454 



429 
404 
378 
353 
117 
30^ 
277 
251 
226 
200 

175 
150 
124 
099 
074 



048 
023 
997 
972 
947 



921 
896 
871 
845 
820 



795 
769 
744 
718 
693 



668 

642 
617 
592 

566 



541 
516 
490 
465 
440 



414 
389 
364 
338 
313 

287 
262 
237 
211 
186 



Log. Cos. 



161 
135 
110 
085 
059 



03 034 



c. d. Log, Tan 



87 107 
87 096 
87 084 
87 073 
87 062 



87 050 
87 039 
87 027 
87 016 
87 004 



86 993 
86 982 
86 970 
86 959 
86 947 



86 938 
86 924 
86 913 
86 901 
86 890 



86 878 
86 867 
86 855 
86 844 
86 832 



86 821 
86 809 
86 798 
86 786 

86 774 



86 763 
86 751 
86 740 
86 728 
86 716 



86 705 
86 693 
86 682 
86 670 
86 658 



86 647 
86 635 
86 623 
86 612 
86 600 



86 588 
86 577 
86 565 
86 553 
86 542 



86 530 
86 518 
86 507 
86 495 
86 483 



86 471 
86 460 
86 448 
86 436 
86 424 



9.86 412 



l^^"" 



Log. Sin. 
58S 



11 
11 
11 
11 

11 
ll 
11 
11 
11 

11 
11 
11 
11 
11 

11 
11 
11 
11 
11 

11 
11 
11 
11 
11 

11 

11 
11 

12 
11 

11 
11 
11 
11 
12 

11 
11 
11 
11 
12 

ll 
ll 
12 
11 
11 

12 
11 
ll 
12 
11 

12 
11 
11 
12 
11 
12 
ll 
12 
11 
12 

12 



d.' 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 
50 
49 
48 
47 
46^ 

45 
44 
43 
42 
41. 

40 
39 
38 
37 
36_ 
35 
34 
33 
32 
3L 

30 
29 
28 
27 
26. 
25 
24 
23 
22 
21 

20 
19 
18 
17 
li 
15 
14 
13 
12 
^1 

10 

9 
8 

7 
_J6 

5 
4 
3 
2 
_1 
O 



P. P. 





25 


25 


6 


2.5 


2-5 


7 


3 





2 


9 


8 


3 


4 


3 


3 


9 


3 


8 


3 


7 


10 


4 


2 


4 


1 


20 


8 


5 


8 


3 


30 


12 


7 


12 


5 


40 


17 


G 


16 


6 


50 


21 


2 


20 


8 





14 


i; 


6 


1.4 


1. 


7 


1 


6 


1. 


8 


1 


8 


1. 


9 


2 


1 


2- 


10 


2 


3 


2. 


20 


4 


6 


4. 


30 


7 





6. 


40 


9 


3 


9. 


50 


11 


6 


11. 



12 


11 




1.2 


1.1 




1 


4 




3 




1 


6 




5 




1 


8 




7 




2 







9 




4 


C 


3 


8 


3. 


6 





5 


7 


5- 


8 





7 


6 


7 


10 





9 


6 


9. 



P. p. 



47 



13' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 136° 



Log. Sin. 



83 378 
83 392 
9-83 405 
9. 83 419 
9. 83 432 



9 . 83 446 
9-83 459 
9-83 473 
9-83 486 
83 500 



9-83 513 
9-83 527 
9-83 540 
9-83 554 
9-83 567 

9 



83 580 
83 594 
83 607 
83 621 
83 634 



9-83 647 
9-83 661 
9-83 674 
9- 83 688 
9-83 701 



9-83 714 
9-83 728 
9-83 741 
9-83 754 
9-83 768 



83 781 
83 794 
83 808 
83 821 
83 834 



83 847 
83 861 
83 874 
83 887 
83 900 



9-83 914 
9-83 927 
9-83 940 
9- 83 953 
9-83 967 



83 980 

83 993 

84 006 
84 019 
84 033 

84 046 
84 059 
84 072 
84 085 
84 098 






9-84 111 
9-84 124 
9-84 138 
9-84 151 
9-84 164 



9-84 177 



Log. Cos, 



13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 
13 
13 
13 
13 
13 

13 



^3^ 



97 092 
97 117 
9-97 143 
9-97 168 
9-97 193 



Log. Tan. 



96 965 
96 991 
9-97 016 
9-97 041 
997 067 



9-97 219 
97 244 
9-97 269 
9-97 295 
9-97 320 



9-97 345 
9-97 370 
9-97 396 
9-97 421 
9-97 446 



97 472 
97 497 
97 522 
97 548 
97 573 



97 598 
97 624 
97 649 
97 674 
97 699 



9-97 725 
9-97 750 
9-97 775 
9-97 801 
9-97 826 



9-97 851 
9 



97 877 
97 902 
97 927 
9-97 952 



97 978 

98 003 
98 028 
98 054 
98 079 



c.d. 



98 104 
98 129 
98 155 
98 180 
98 205 



9-98 231 
9-98 256 
9-98 281 
9-98 306 
9-98 332 



9-98 357 
9-98 382 
9-98 408 
9-98 433 
9. 98 458 



9-98 483 
Log. Cot. 



25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
C.d 



0-03 034 
003 009 
02 984 
0-02 958 
0-02 933 



0-02 908 
0-02 882 
02 857 
02 832 
02 806 



Log. Cot, 



02 781 
02 756 
0-02 730 
0-02 705 
0-02 68C 



02 654 
02 629 
0-02 604 
0-02 578 
0-02 553 



Log. Cos. 



86 412 
9-86 401 

86 389 
9-86 377 
9-86 365 



86 354 
86 342 
86 330 
86 318 
86 306 



-86 294 
-86 282 
-86 271 
-86 259 
-86 247 



0-02 528 
0-02 502 
0.02 477 
0-02 452 
0-02 427 



02 401 
.02 376 
-02 351 
-02 325 

02 300 



0-02 275 
0-02 249 
0-02 224 
0-02 199 
0-02 174 



- 02 148 
0-02 123 
0-02 098 





0-02 022 
0-01 996 
0-01 971 
0-01 946 
0-01 921 



02 072 
02 047 



0-01 895 
0-01 87C 
0-01 845 
0-01 819 
001 794 



0-01 769 
0-01 744 
0-01 718 
0-01 693 
0-01 668 



0-01 64? 
0-01 617 
0.01 592 
0-01 567 
0-01 541 



0-01 51^ 
Log. Tan 



86 235 
86 223 
86 211 
86 199 
86 187 



86 176 
86 164 
86 152 
86 140 
86 128 



9- 86 116 
86 104 
9-86 092 
9-86 080 
9-86 068 



9-86 056 
9-86 044 
9-86 032 
9-86 020 
9- 86 008 



9-85 996 
9-85 984 
9-85 972 
9-85 960 
P. 85 948 



85 936 

85 924 

-85 912 

• 85 900 

-85 887 



9-85 875 
9-85 863 
9-85 851 
9-85 839 
9-85 827 



9-85 815 
9-85 803 
9-85 791 
9-85 778 
9-85 766 



9-85 754 
9-85 742 
9- 85 730 
9-85 718 
9-85 705 



9-85 693 
Log. Sin. 



11 
12 
11 
12 

ll 

12 
12 
11 
12 

12 
12 
11 
12 
12 

11 
12 
12 
12 
12 

11 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 
12 
12 
12 
12 

12 



5S9 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46 

45 
44 
43 
42 

40 

39 
38 
37 
36 

35 

34 
33 
32 

-31 
30 

29 

28 

27 

-26 

25 
24 
23 
22 
_2]_ 

20 

19 
18 

17 
16 

15 
14 
13 
12 
11 

10 

9 
8 

7 
6 

5 
4 
3 
2 
1 
O 



P. P. 



25 



2 


5 


2. 


3 





2. 


3 


4 


3. 


3 


8 


3. 


4 


2 


4. 


8 


5 


8 


12 


7 


12 


17 





16 


21 


2 


20. 



25 

5 
9 
3 
7 
1 
3 
5 
6 
8 





13 


13 


6 


1.3 


1.3 


7 


1 


6 


1 


5 


8 


1 


8 


1 




9 


2 





1 


9 


10 


2 


2 


2 


1 


20 


4 


5 


4 


3 


30 


6 


7 


6 


5 


40 


9 





8 


6 


50 


11 


2 


10 


8 





12 


12 


11 


6 


1.2 


1.2 


1.1 


7 


1 


4 


1 


4 




3 


8 


1 


6 


1 


6 


1 


5 


9 


1 


9 


1 


8 




7 


10 


2 


1 


2 







9 


20 


4 


1 


4 


03 


8 


30 


6 


2 


6 


5 


7 


40 


8 


3 


8 


7 


6 


50 


10 


4 


10 





9 


6 



P. p. 



46^ 



44° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



135' 





1 

2 

3 

j4 

5 
6 
7 
8 
_9^ 

10 

11 
12 
13 

H 

15 
16 
17 
18 

ii. 

20 

21 

22 

23 

21 

25 

26 

27 

28 

21 

30 

31 

32 

33 

3£ 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
4£ 

45 
46 
47 
48 
4SL 

50 

51 
52 
53 
5£ 

55 
56 
57 
58 
59^ 

60 



Log. Sin, 



84 177 
84 190 
84 203 
84 216 
84 229 



84 242 
84 255 
84 288 
84 281 
84 294 



84 307 
84 320 
84 338 
84 346 
84 359 



84 372 
84 385 
84 398 
84 411 
84 424 



84 437 
84 450 
84 463 
84 476 
84 489 



84 502 
84 514 
84 527 
84 540 
84 553 



84 566 
84 579 
84 592 
84 604 
84 617 



84 630 
84 643 
84 656 
84 669 
84 681 



84 694 
84 707 
84 720 
84 732 
84 745 



84 758 
84 771 
84 783 
84 796 
84 809 



84 822 
84 834 
84 847 
84 860 
84 872 



84 885 
84 898 
84 910 
84 923 
84 936 



9-84 948 



Log. Cos. 



d. 

13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

12 
13 
13 
13 

13 

13 
12 
13 
13 
13 
12 
13 
13 
12 
13 

13 
12 
13 
13 
12 

13 
12 
13 
12 
13 
12 
13 
12 
13 
12 

13 
12 
12 
13 
12 

12 
13 
12 
12 
13 
12 



d. 



Log. Tan. 



9.98 
98 
98 
98 
98 



483 
509 
534 
559 
585 



610 
635 
860 
686 
711 



736 
762 
787 
812 
837 



863 
888 
913 
938 
984 



989 
014 
040 
065 
090 

115 
141 
168 
191 
216 

242 
267 
292 
318 
343 



368 
393 
419 

444 
4B9 



494 
520 
545 
570 
595 



621 
646 
871 
697 

722 



747 
772 
798 
823 
848 



873 
899 

924 
949 
974 



0.00 000 



Log. Cot. c.d 



c.d 

25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 

25 
25 
25 
25 

25 
25 
25 
25 
25 

25 

25 
25 
25 
25 

25 

25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 



Log. Cot. 



01 516 
01 491 
01 465 
01 440 
01 415 



01 390 
01 384 
01 339 
01 314 
01 289 



01 263 
01 238 
01 213 
01 187 
01 162 



01 137 
01 112 
01 086 
01 061 
01 036 



01 010 
00 985 
00 960 
00 935 
00 909 



00 884 
00 859 
00 834 
00 808 
00 783 



00 758 
00 733 
00 707 
00 682 
00 657 



00 631 
00 606 
00 581 
00 556 
00 530 



00 505 
00 480 
00 455 
00 429 
00 404 



00 379 
00 353 
00 328 
00 303 
00 278 



00 252 
00 227 
00 202 
00 177 
00 151 



00 126 
00 101 
00 076 
00 050 
00 025 



0.00 000 



Log. Tan, 



Log. Cos. 



85 693 
85 681 
85 669 
85 657 
85 644 



85 632 
85 620 
85 608 
85 595 
85 583 



85 571 
85 559 
85 546 
85 534 
85 522 



85 509 
85 497 
85 485 
85 472 
85 460 



85 448 
85 435 
85 423 
85 411 
85 398 



85 386 
85 374 
85 361 
85 349 
85 336 



85 324 
85 312 
85 299 
85 287 
85 274 



85 262 
85 249 
85 237 
85 224 
85 212 



85 199 
85 187 
85 174 
85 162 
85 149 



85 137 
85 124 
85 112 
85 099 
85 087 



85 074 
85 062 
85 049 
85 037 
85 024 



85 Oil 
84 999 
84 986 
84 974 
84 961 



9. 84 948 



134' 



Log. Sin, 
590 



d. 



d. 



60 

59 
58 
57 
5Q 

55 
54 
53 
52 
11 
50 
49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
31 
35 
34 
33 
32 
31 
30 
29 
28 
27 
21 
25 
24 
23 
22 

21 
20 

19 
18 
17 
11 
15 
14 
13 
12 

n 

10 

9 
8 

7 
_6^ 
5 
4 
3 
2 

o 



p. p. 





25 


25 


6 


2.5 


2.5 


7 


3 





2 


9 


8 


3 


4 


3 


3 


9 


3 


8 


3 


7 


10 


4 


2 


4 


1 


20 


8 


5 


8 


3 


30 


12 


7 


12 


5 


40 


17 





16 


6 


50 


21 


2 


20 


8 



13 



1 


3 


1. 


1 


6 


1. 


1 


8 


1. 


2 





1. 


2 


2 


2. 


4 


5 


4. 


6 


7 


6. 


9 





8. 


11 


2 


10. 



13 

3 
5 

7 
9 
I 
3 
5 



12 



1 


2 


1 


4 


1 


6 


1 


9 


2 


1 


4 


1 


6 


2 


8 


3 


10 


4 



12 

1.2 



10.0 



P.P. 



TABLE VIII 



LOGARITHMIC VERSED SINES AND EXTERNAL 

SECANTS. 

591 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL 
0° SECANTS. 1° 



Log. Vers. 



62642 
22848 
53066 
83054 



02436 
18272 
31632 
43230 
53490 



62342 
70920 
78478 
85431 
91888 



97880 
03466 
08732 
13696 
18393 



22848 
27086 
31126 
34987 
38684 



42230 
45636 
48915 
52073 
55121 



58086 
60914 
63672 
66344 
68937 



71455 
73902 
76282 
78598 
80854 



83053 
85198 
87291 
89335 
91332 



93284 
95193 
97061 
98890 
00680 



02435 
04155 
05842 
07496 
09120 



10714 
12279 
13816 
15327 
16811 



6. 18271 



Log, Vers, 



60206 
35218 
24987 
19382 
15836 
13389 
11598 
10230 

9151 
8278 
7558 
6953 
6437 

5992 
5605 
5266 
4964 
4696 

4455 
4238 
4040 
3861 
3697 

3545 
3406 
3278 
3158 
3048 

2944 
2848 
2757 
2672 
2593 

2518 
2447 
2379 
2316 
2256 

2199 
2145 
2093 
2044 
1996 
1952 
1909 
1868 
1829 
1790 

1755 
1720 
1686 
1654 
1623 

1594 
1565 
1537 
1511 
1484 

1460 



Log. Exsec, 



— 00 

62642 
22848 
58066 
83054 



02436 
18272 
31662 
43260 
53491 



62642 
70921 
78478 
85431 
91868 



97861 
03466 
08732 
13697 
18393 



22849 
27087 
31127 
34988 
38685 



42231 
45638 
48916 
52075 
55123 



58068 
60916 
63674 
66346 
68940 



71457 
73904 
76284 
78601 
80857 



83056 
85201 
87295 
89338 
91335 



93288 
95197 
97065 
98894 
00685 



02440 
04160 
05847 
07501 
09125 



10719 
12284 
13822 
15333 
16818^ 

6. 18278 



Log. Exsec. 



60206 
35218 
24987 
19382 
15836 
13389 
11598 
10230 

9151 
8279 
7557 
6952 
6437 

5993 
5605 
5286 
4964 
4696 

4456 
4238 
4040 
3861 
3697 

3545 
3407 
3278 
3159 
3048 

2945 
2848 
2758 
2672 
2593 

2517 
2447 
2380 
2316 
2256 

2199 
2145 
2093 
2043 
1997 

1952 
1909 
1868 
1829 
1791 

1755 
1720 
168Z 
1654 
1623 
1594 
1565 
1537 
1511 
1485 
1460 



1> 



Log. Vers. 



6.18271 
.19707 
.21119 
.22509 
•23877 



6.25223 
.26549 
•27856 
.29142 
•30410 



6.31660 
•32892 
•34107 
•35305 
•36487 



6.37653 
.38803 
•39938 
•41059 
•42165 



6^43258 
•44337 
•45403 
•46455 
.47496 



6-48524 
.49539 
•50544 
.51536 
•52518 



6.53488 
. 54448 
.55397 
.56336 
.57265 



6-58184 
•59093 
.59993 
.60884 
.61766 



6.62639 
.63503 
.64359 
.65206 
.66045 



6.66876 
.67700 
.68515 
-69323 
•70124 



6.70917 
-71703 
.72482 
.73254 
•74019 



8-74777 
-75529 
-76275 
•77014 
-77747 



fi- 78474 



Log. Vers. 



2> 



1435 
1412 
1389 
1368 
1346 
1326 
1306 
1286 
1268 

1250 

1232 

1214 

1198 

1182 

1166 

1150 

1135 

1121 

1106 

1093 

1078 

1066 

1052 

1040 

1028 

1016 

1004 

992 

981 

970 

960 

949 

939 

929 

919 
909 
900 
891 
882 
872 
864 
855 
847 
839 

831 
823 
815 
808 
800 

793 
786 
779 
772 
765 

758 
752 
745 
739 
733 
726 



JD 



Log. Exsec. 



6.18278 
19714 
21126 
22516 
23884 



25231 
26557 
27864 
29151 
30419 



31669 
32901 
34116 
35315 
36_497^ 
37663 
38814 
39949 
41070 
42177 



43270 
44349 
45415 
46468 
47509 



48537 
49553 
50557 
51550 

52532 



503 

'4463 

55413 

56352 

57281 



58201 
59110 
60011 
60902 
61784 



62857 
63522 
84378 
65226 
66065 



66897 
67720 
68536 
69345 
70145 



70939 
71725 
72505 
73277 
74043 



74802 
75554 
76300 
77040 
77773 



78500 



Log, Exsec, 



592 



TABLE VIII.— LOGARITHMIC VEilSi:D SINES AND EXTERNAL 
3° SECANTS. 3° 



' Log. Vers. 



78474 
79195 
79909 
80618 
8132|. 
82019 
82711 
83398 
84079 
84755_ 

85425 
86091 
86751 
87407 
88057 



88703 
89344 
89980 
90612 
91239 



91862 
92480 
93093 
93703 
94308 



94909 
95506 
96099 
96688 
97272 

97853 
98430 
99004 
99573 
00139 



00701 
01259 
01814 
02366 
0^914 



03458 
03999 
04537 
05071 
056^3 



06130 
06655 
07177 
07695 
08211 



08723 
09232 
09739 
10242 
10743 



11240 
11735 
12227 
12716 
13203 



13687 



Log. Vers. 



721 
714 
709 
703 

697 
692 
688 
681 
676 

670 
665 
660 
655 
650 

646 
641 
636 
631 
627 
622 
618 
613 
609 
605 

601 
597 
592 
539 
584 

581 
577 
573 
569 
535 

562 
558 
555 
551 
548 

544 
541 
537 
534 
53l 

527 
525 
521 
518 
515 

51^ 
505 
508 
503 
500 

497 
495 
492 
489 
486 

484 



2> 



Log. Exsec. 



78500 
79221 
79937 
80646 
81350 

82048 
82740 
83427 
84109 
84785 



85457 
861^3 
86783 
87439 
88090 



88737 
89378 
90015 
90647 
91275 



91898 
92516 
93131 
93741 
94346 

94948 
95545 
96139 
96728 
97313 

97895 
98472 
99046 
99616 
00182 



00745 
01304 
01860 
02412 
02960 



03505 
04047 
04585 
05120 
05652 



06180 
06706 
07228 
07747 
08263 



08776 
09286 
09793 
10297 
10798 



11297 
11792 
12285 
12775 
13262 

13746 



n 



721 
715 
709 
703 

698 
692 
687 
682 
676 

671 
666 
660 
656 
651 

646 
641 
630 
632 
628 

623 
618 
614 
610 
605 

601 
597 
593 
589 
585 

581 
577 
574 
570 
566 

563 
559 
555 
552 
548 

545 
541 
538 
535 
53l 

528 
525 
522 
519 
516 
513 
509 
507 
503 
501 

498 
495 
493 
490 
487 
484 



Log. Exsec. I I> 



Log. Vers, 



13687 
14168 
14646 
15122 
15595 



16066 
16534 
17000 
17463 
17923 
18382 
18837 
19291 
19742 
20191 

20637 
21081 
21523 
21863 
22400 



22836 
23269 
23700 
24129 
24555 



24980 
25402 
25823 
26241 
26658 



27072 
27485 
27895 
28304 
28711 



29116 
29518 
29919 
30319 
30716 



31112 
31505 
31897 
32288 
32676 



33063 
33448 
33831 
34213 
34593 



34971 
35348 
35723 
36097 
3646§ 



36839 
37207 
37574 
37940 
38304 



38fifi7 



Log. Vers. 



D 



481 
478 
475 
473 

470 
468 
466 
463 
460 

458 
455 
453 
451 
448 

446 
444 
442 
440 
437 

435 
433 
431 
429 
426 

424 
422 
420 
418 
416 

414 
412 
410 
409 
406 

405 
402 
401 
399 
397 

395 
393 
392 
390 
388 

386 
385 
383 
382 
380 

378 
377 
375 
373 
37l 
370 
368 
367 
368 
364 

362 



2) 



Log, Exsec, 



7-13746 
14228 
14707 
15183 
15657^ 

16129 
16598 
17064 
17528 
1_7989 

18448 
18905 
19359 
19811 
20260 



20707 
21152 
21595 
22035 
22473 



22909 
23343 
23775 
24204 
24632 



25057 
25480 
25902 
26321 
26738 



27153 
27567 
27978 
28387 
28795 



29200 
29604 
30006 
30406 
30804 



31201 
31595 
31988 
32379 
32768 



33156 
33542 
33926 
34309 
34689 



35069 
35446 
358^2 
36196 
36569 



36940 
37310 
37678 
38044 
38409 



7-38773 



Log. Exsec. 



481 
479 
476 
474 

471 
469 
466 
464 
46l 

459 
456 
454 
452 
449 

447 
445 
442 
440 
438 

436 
434 
431 
429 
427 
425 
423 
421 
419 
417 
415 
413 
411 
409 
407 

405 
404 
402 
400 
398 

396 
394 
393 
391 
389 

388 
385 
384 
382 
380 

379 
377 
376 
374 
373 

371 
369 
368 
366 
365 

363 



X> 



593 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
4° 5° 



O 

1 

2 

3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 



30 

21 
22 
23 
2£ 

25 

26 

27 

28 

29_ 

30 

31 

32 

33 

3£ 

35 
36 
37 
38 
39 



40 

41 
42 
43 
4£ 

45 

46 

47 

48 

41 

50 

51 

52 

53 

51 

55 
56 
57 
58 
59_ 
60 



Lg. Vers. 



38667 
39028 
39387 
39745 
40102 



40457 
40810 
41163 
41513 
41863 



42211 
42557 
42903 
43246 
43589 



43930 
44270 
44608 
44946 
45281 



45616 
45949 
46281 
46612 
46941 



47270 
47597 
47922 
48247 
48570 



48892 

49213 
49533 
49852 
50169 



50485 
50800 
51114 
51427 
51739 



52050 
52359 
52667 
52975 
53281 



53586 
53890 
54193 
54495 
54796 



55096 
55395 
55692 
55989 
56285 



56580 
56873 
57166 
57458 
57749 



7. 58039 



Lg. Vers, 



n 



361 
359 
358 
356 

355 
353 
352 
350 

349 

348 

346 
345 
343 
342 

341 
339 
338 
337 
335 

334 
333 
332 
330 
329 
328 
327 
325 
324 
323 

322 
321 
320 
318 
317 
316 
315 
314 
313 
311 

311 
309 
308 
307 
306 

305 
304 
303 
302 
300 

300 
299 
297 
297 
295 

295 
293 
293 
292 
290 

290 



Log.'Exs, 



38773 
39134 
39495 
39854 
40211 



40567 
40922 
41275 
41627 
41977 



42673 
43019 
43364 
43708 



44050 
44390 
44730 
45068 
45405 



45740 
46075 
46407 
46739 
47070 



47399 
47727 
48054 
48379 
48703 



49026 
49348 
49669 
49989 
50307 



50624 
50941 
51256 
51569 
51882 



52194 
52504 
52814 
53122 
53429 



53735 
54041 
54345 
54648 
54950 



55251 
55550 
55849 
56147 
56444 



56740 
57035 
57329 
57621 
57913 



58204 



J^ Log. Exs, 



J> 



361 
360 
359 
357 
356 
354 
353 
352 
350 

349 
347 
346 
345 
343 

342 
340 
339 
338 
337 

335 
334 
332 
332 
330 

329 
328 
327 
325 
324 

323 
322 
321 
319 
318 

317 
316 
315 
313 
313 

311 
31C 
309 
308 
307 

306 
305 
304 
303 
302 

301 
299 
299 
298 
296 

296 
295 
294 
292 
292 

291 



2> 



Lg. Vers, 



58039 
58328 
58615 
58902 
59188 



59473 
59758 
60041 
60323 
60604 



60885 
81164 
61443 
61721 
61998 



62274 
62549 
62823 
63096 
63369 



63641 
63911 
64181 
64451 
64719 



64986 
65253 
65519 
65784 
66048 



66311 
66574 
66836 
67097 
67357 



67617 
67875 
68133 
68390 
68647 



68802 
69157 
69411 
69665 
69917 



70169 
70421 
70671 
70921 
71170 



71418 
71666 
71913 
72159 
72404 



72649 
72893 
73137 
73379 
73621 



73863 



Lg. Vers. 



D 



289 

287 
287 
286 

285 
284 
283 

282 
281 

280 

279 
279 
277 
277 

276 

275 
274 
273 
272 

272 
270 
27C 
269 
268 

267 
266 
266 
265 
264 

263 
263 
261 
261 
260 

259 
258 
258 
257 
256 

255 
255 
254 
253 
252 

252 
251 
250 
25C 
249 

248 
247 
247 
246 
245 

245 
244 
243 
242 
242 

241 



Log. Exs, 



58204 
58494 
58783 
59071 
59358 



59645 
59930 
60214 
60498 
6078C 



61062 
61342 
61622 
61901 
62179 



62456 
62733 
63008 
63282 
63556 



63829 
64101 
64372 
64643 
64912 



65181 
65449 
65716 
65982 
66247 



66512 
66776 
67039 
67301 
67562 



67823 
68083 
68342 
68601 
68858 

69115 
69371 
69627 
69881 
70135 



70388 
70641 
70893 
71144 
71394 



71644 
71892 
72141 
72388 
72635 



72881 
73126 
73371 
73615 
73859 



74101 



Log. Exs. 



D 



290 
289 
288 

287 

286 
285 
284 
283 
282 

28l 
280 
28C 
27S 
278 

277 
276 
275 
274 
274 

273 
272 
271 
270 
269 

269 
268 
267 
266 
265 
264 
264 
263 
262 
261 

261 
260 
259 
258 
257 

257 
256 
255 
254 
254 

253 
252 
252 
251 
25C 

25C 
248 
248 
247 
246 

246 
24^5 
245 
244 
243 
242 



Z> 



P.P. 



10 

11 

12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
j49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
_5i 

60 





36 





350 


6 


360 


35. 01 


7 


42 





40 


8 


8 


48 





46 


6 


9 


54 





51 


5 


10 


60 





58 


3 


20 


120 





116 


6 


30 


180 





175 





40 


240 





233 


3 


50 


300 





291 


6 



340 

34. 

39.6 

45-3 

51.0 

56.6 

113.3 

170-0 

226.6 

283. 3 





330 


320 


310 


6 


33. 


32.0 


31.0 


7 


38.5 


37 


3 


36 


1 


8 


44-0 


42 


6 


41 


3 


9 


49.5 


48 





46 


5 


10 


55.0 


53 


3 


51 


6 


20 


110.0 


106 


6 


103 


3 


30 


165.0 


160 





155 





40 


220.0 


213 


3 


206 


6 


50 


275.0 


266 


6 


258 


3 





300 


290 


28( 


6 


30.0 


29.0 


28. 


7 


35 





33 


8 


32. 


8 


40 





38 


6 


37. 


9 


45 





43 


5 


42. 


10 


50 





48 


3 


46. 


20 


100 





96 


6 


93. 


30 


150 





145 





140. 


40 


200 





193 


3 


186. 


50 


250 





241 


6 


233. 





270 


260 


25 


6 


27.0 


26.0 


25. 


7 


31 


5 


30 


3 


29. 


8 


36 





34 


6 


33. 


9 


40 


5 


39 





37. 


10 


45 





43 


3 


41. 


20 


90 





86 


6 


83. 


30 


135 





130 





125. 


40 


180 





173 


3 


166. 


50 


225 





216 


6 


208. 





240 


230 


22( 


6 


24.0 


23.0 


22. 


7 


28 





26 


8 


25- 


8 


32 





30 


6 


29. 


9 


36 





34 


5 


33. 


10 


40 





38 


3 


36. 


20 


80 





76 


6 


73. 


30 


120 





115 





110. 


40 


160 





153 


3 


146- 


50 


200 





191 


6 


183. 





210 


200 


190 


6 


21.0 


20.0 


19. 


7 


24 


5 


23 


3 


22 


1 


8 


28 





26 


g 


25 


3 


9 


31 


5 


30 





28 


5 


10 


35 





33 


3 


31 


6 


20 


70 





66 


6 


63 


3 


30 


105 





100 





95 





40 


140 





133 


3 


126 


6 


50 


175 





166 


6 


158 


3 



P. p. 



594 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTJ 
i 6° 7° 



Lg. Vers. -O Log.Exs. X>;Lg. Vers. 



73863 
74104 
74344 
74583 
74822 



75060 
75297 
75534 
75770 
76006 



76240 
76475 
76708 
76941 
77173 



77405 
77636 
77867 
78097 
78326 



78554 
78783 
79010 
79237 
79463 



79689 
79914 
80138 
80362 
80586 



80808 
81031 
81252 
81473 
81694 

81914 
82133 
82352 
82570 
82788 



83005 
83222 
83438 
83653 
83868 



84083 
84297 
84510 
84723 
84935 



85147 
85359 
85570 
85780 
85990 



88199 
86408 
88616 
86824 
87031 



7-87238 



Lg. Vers 



241 
240 
239 
239 

238 

237 
236 
236 
235 

234 
234 
233 
233 
232 

232 
231 
230 
230 

229 

228 
228 
227 
227 
226 

225 
225 

224 
224 
223 

222 
222 
221 
221 
220 
220 
219 
219 
218 
217 
217 
217 
216 
215 
215 

214 
214 
213 
213 
212 

212 
211 
211 
210 
210 

209 
209 
208 
208 
207 
206 



7-74101 
74343 
74585 
74826 
75066 



75305 
75544 
75782 
76019 
76256 



76492 
76728 
76963 
77197 
77431 



77664 
77897 
78128 
78360 
78590 



78820 
79050 
79279 
79507 
79735 



79962 
80188 
80414 
80639 
80864 



81088 
81312 
81535 
81758 
81980 



82201 
82422 
82642 
82832 



83300 
83518 
83735 
83952 
84169 



84385 
84600 
84815 
85030 
85243 



85457 
85670 
85882 
86094 
86305 



86516 
86726 
86936 
87146 
87354 



875ft3 



242 
24l 
241 
240 
239 
239 
238 
237 
237 

236 
235 
235 
234 
233 

233 
232 
231 
231 
230 

230 
229 
229 
228 
228 

227 
226 
226 
225 
225 

224 
224 
223 
222 
222 
221 
221 
220 
219 
219 

219 
218 
217 
217 
216 

216 
215 
215 
214 
213 

213 
213 
212 
211 
211 

211 
210 
210 
209 
208 

208 



Log.Exs. 



n 



87238 
87444 
87650 
87855 
88060 



88264 
88468 
88672 
88875 
19PJL7 
89279 
89481 
89682 
89882 
90082 



90282 
90481 
90680 
90878 
91076 



91273 
91470 
91667 
91863 
92058 



92253 
92448 
92642 
92836 
93029 



93222 
93415 
93307 
93799 
93990 



94181 
94371 
94561 
94751 
94940 



95129 
95317 
95505 
95693 
95880 



98066 
98253 
96439 
96624 
98809 



96994 
97178 
97362 
97546 
97729 



97912 
98094 
98276 
98458 
98639 



98820 



Lg. Vers 



206 
205 
2u5 
204 

204 
204 
203 
203 
202 

202 
201 
201 
200 
200 

199 
199 
198 
198 
197 

197 
197 
196 
196 
195 

195 
195 
194 
194 
193 

193 
192 
192 
191 
191 

190 
190 
190 
189 
189 

189 
188 
187 
188 
187 

186 
186 
186 
185 
185 
184 
184 
184 
183 
183 
183 
182 
182 
182 
181 

181 



Log. Exs. /> 



87563 
87771 
87978 
88185 
88391 



88597 
88803 
89008 
89212 
89416 



89620 
89823 
90025 
90228 
90429 



90630 
9083i 
91032 
91231 
91431 



91630 
91828 
92027 
92224 
92421 



92618 
92815 
93010 
93206 
93401 



93596 
93790 
93984 
94177 
94370 



94562 
94754 
94946 
95137 
95328 



95519 
95709 
95898 
96088 

96276 



98465 
96653 
96841 
97028 
97215 



97401 
97587 
97773 
97958 
98143 



98327 
98512 
98695 
98879 
99062 



99244 



I> Log.Exs, 
595 



208 
207 
207 
206 

206 
205 
205 
204 
204 

203 
203 
202 
2C2 
201 

201 
201 
200 
199 
199 

199 
198 
198 
197 
197 

197 
196 
195 
195 
195 

195 
194 
194 
193 
193 
192 
192 
192 
191 
191 

190 
190 
189 
189 
188 

188 
188 
188 
187 
187 

186 
186 
185 
185 
184 

184 
184 
183 
183 
183 
182 



n 



10 

11 

12 
13 

]A 

15 
16 
17 
18 

11 
20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
J4_ 

45 
46 
47 
48 
49 



50 

51 
52 
53 
54 

55 
56 
57 
58 

00 



P.P. 



180 9_ 

9 
1 
2 
4 
6 
1 

2 

3 
9 



6 


18 








7 


21 





1 


8 


24 





1 


9 


27 





1 


10 


30 





1 


20 


60 





3 


30 


90 


C 


A 


40 


120 


C 


3 


5C 


150 


I 


7 



9 

0.9 



0.8 





8 


0. 


1.0 





9 


0. 


1-1 


1 





1. 


1-3 


1 


2 


1. 


1-4 


1 




1. 


2-8 


2 


g 


2. 


4-2 


4 





3. 


5-6 


5 


3 


5. 


7.1 


16 


6 


6. 





7 


6 


( 


6 


0.7 


0.6 





7 


0.8 


0.7 





8 


0.9 


0.8 


0. 


9 


1.0 


1-0 





10 


1.1 


1.1 


1. 


20 


2.3 


2.1 


2. 


30 


3-5 


3.2 


3. 


40 


4.6 


4.3 


4. 


50 


5-8 


5-4 


5. 





5 


5 




4 


6 


0.5 


0.5 


0.41 


7 


0.6 





6 





5 


8 


0.7 





6 





6 


9 


08 










7 


10 


9 





8 





7 


20 


1.8 


1 


6 


1 


5 


30 


2 = 7 


3 


5 


2 


2 


40 


3.6 


3 




3 





50 


46 


4 


1 


3 


7 



6 
-7 

8 

9 




• 





4 

0-4 
0.4 
0.5 
0.6 
0.6 
1.3 
2.0 
2.6 
3-3 





3 


3 




3 


2 


6 


0.3 


0.3 


0.210.2 


7 


0.4 





3 





3,0 


2 


8 


0.4 





4 





30 


2 


9 


0.5 





4 





4 


3 


10 


0.6 





5 





40 


3 


20 


1.1 


1 


C 





80 


6 


30 


1.7 


1 


5 


1 


21 





40 


2.3 


2 





1 


61 


3 


50 


2.9 


2 


5 


2 


111 


6 



1_ 

60. 1 
7i0.2 
8,0.2 
9 0.2 
100.2 

2o;o 5 

30,0.f 
40,1.0 
50 1.5 



1 


( 


f> 


0.1 


0.0 
















" 











1 





1 





1 





1 





3 





1 





5 





2 





6 





3 





8 





4 



P.P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
8° 9° 



Lg. Vers, 



O 

1 

2 

3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 

il 
20 

21 
22 
23 
24 

25 

26 

27 

28 

2i 

30 

31 

32 

33 

34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



98820 
99000 
99180 
99360 
99539 



99718 
99897 
00075 
00253 
00431 



Log.Exs, 



00608 
00784 
00961 
01137 
01313 



01488 
01663 
01838 
02012 
02186 



7-99244 
.99427 
.9960? 
.99790 

7. 99971 



02359 
02533 
02706 
02878 
03050 



03222 
03394 
03565 
03736 
03906 



04076 
04246 
04416 
04585 
04754 



04922 
05090 
05258 
05426 
05593 



05780 
05926 
06093 
06259 
06424 



06589 
06754 
06919 
07083 
07247 



07411 
07575 
07738 
07900 
08063 



08225 
08387 
08549 
08710 
08871 



8.00152 
.00332 
.00512 
.00692 
.00871 



J> 

180 
180 
179 
179 
179 
178 
178 
177 
178 

177 
176 
176 
176 
176 

175 
175 
175 
174 
174 

1 7^ 

17^ 8.02820 
^'^ .02995 
.03170 
.03345 
•03519 

8.03692 
.03866 
.04039 
.04212 
•04384 



n 



.01050 
.01229 
.01407 
.01585 
.01763 



8-01940 
.02117 
.02293 
.02469 
.02645 



182 
182 
181 
181 
180 
180 
180 
180 
179 
179 
178 
178 
178 
177 
177 
177 
176 



09031 



jLg. Vers. 



173 
172 
172 
172 
171 
171 
171 
170 
170 
170 
169 
169 
169 

168 
168 
168 
167 
167 

167 
166 
166 
166 
165 

165 
165 
165 
164 
164 

164 
163 
163 
162 
162 
162 
161 
162 
161 
161 
160 

1} 



•04556 
.04728 
.04899 
.05070 
•05241 



Lg. Vers. 



•09031 
.09192 
.09352 
.09512 
•09671 



.09830 
•09989 
.10148 
.10306 
.10464 



8.10622 
.10779 
.10936 
.11093 
.11250 



8.11406 
.11562 
.11718 
.11873 

.19029 

175 r 



176 
175 



^'l 8.12184 

^'^ .12338 

.12492... 
.12647 IrI 
• 12800 -^^f 

8.12954 ]ll 
.13107 ,to 
.13260 
.13413 
•13565 



•05411 
.05581 
.05751 
.05921 
.06090 



8.0625? 
.06427 
.06595 
.06763 
•06931 



807098 
•07265 
.07431 
.07598 
• 077*64 



8-07929 
.08095 
.08260 
.08424 
-08589 



•08753 
-08917 
.09081 
.09244 
.09407 



809569 



175 
174 
174 

173 
173 
173 
173 
172 

172 
171 
17l 
171 
170 

170 
170 
170 
169 
169 

169 
168 
168 
168 
167 

167 
167 
166 
166 
166 

165 
165 
165 
164 
164 

164 
163 
164 
163 
163 
162 



Log.Exs, 



JJ 



2> 

160 
160 
160 
159 
159 
159 
158 
158 
158 

157 
157 
157 
157 
156 

156 
156 
155 
155 
155 

155 
154 
154 



Log.Exs. 



8^09569 
.09732 
.09894 
.10056 
•10217 



8-13717 
.13869 
.14021 
.14172 
-14323 



. 14474 
.14625 
.14775 
.14925 
• 15075 



15225 
15374 
15523 
15672 
15820 



8-15968 
.16116 
.16264 
.16412 
-16559 



1-16706 
.16852 
.1699? 
.17145 
.17291 



8-17437 
-17582 
.17728,45 
.17873 {II 
-18017 ^*! 

8.18162 ^^' 



Lg. Vers, 



153 
152 
152 

152 
152 
151 
151 
151 

151 

150 
150 
150 
149 

150 
149 
149 
14? 
148 
148 
148 
148 
147 
147 

147 
146 
146 
146 
146 
145 
145 
145 



n 

162 
162 
162 
161 

161 
161 
160 
160 
160 

.11180 1^^ 
.11340 j^S 
.11499 }ll 



8-10378 
-10539 
-10700 
.10860 
-11020 



11658 



159 



•12762 
•1291? 
.13075 
.13232 
•13387 



11816 1^? 

11975 ]ll 
.12133 t?o 
.12291 t^l 

12448 \l] 

157 
157 
156 
156 
155 
156 
155 
155 
154 
154 

154 
154 
153 
153 
153 

153 
152 



.13543 
.13698 
.13854 
.14008 
•14163 



8-14317 
. 14471 
.14625 
.14778 
•14932 



•15085 
.15237 
.15390 
.15542 
. 1 f^694 



15846 
15997 
16148 
16299 
16450 



•16600 
■16750 

16900 
.17050 

17199 



8.1734? 
.17497 
.17646 
.17795 
.17943 



•18091 
.18238 
.18386 



18533 



•18680 



l> Log.Exs 
596 



152 
152 
152 

15? 
15] 
151 
151 
15C 

15C 
150 
150 
149 
149 
14? 
148 
14? 
148 
148 

148 
147 
147 
147 



^4? 



10 

11 
12 
13 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 

32 

33 

J4 

35 
36 
37 
38 
39 

40 

41 
42 
43 
j44 

45 
46 
47 
48 
49 

50 

51 
52 
53 

54 

55 
56 
57 
58 



1 5_9 

18827 ^^^m 



p.p. 



6 
7 
8 
9 

10 
20 
30 
40 
50 



180 

18 

21 

24 

27 

SO 

60 

90 
120 
150 



170 

17^0 
19 



22 
25 
28 
56 
85 
113 
141 



160 

16-0 
18 



21 
24 
26 
53 
80 
1C6 
1S3 



6 

7 
8 
9 
10 
20 
30 
40 
50 



150 

15^0 

17 

20 

22 

25 

50 

75 
100 
125 



140 

14^0 
16 



18 

21 
23 
46 
70 
93 
116 





9 


9 


6 


0^9 


0-9 


7 


1-1 


1-0 


8 


1.2 


1.2 


9 


1-4 


1-3 


10 


1-6 


1-5 


20 


3-1 


3-0 


30 


4-7 


4-5 


40 


6-3 


6-0 


50 


7.9 


7-5 



8 
0-8 
1-0 
l-I 
1-3 
1-4 
2-8 
4-2 
5^6 
I7.I 



6 
7 
8 
9 

10 
20 
30 
40 
50 



8 

0-8 

1 
1 
1 
2 
4 
5 
6 



7_ 
0^7 
09 
1.0 

11 
1.2 
2^5 
3^7 
50 
6.2 



6 

7 

8 

9 

10 

20 

80 

40 

50 






6 


0^ 





7 


0^ 





8 


0^ 


1 





0^ 


1 


1 


1- 


2 


1 


2- 


3 


2 


3- 


4 


3 


4:. 


5 


4 


5. 






5 


0^ 





6 


0^ 





7 


0^ 





8 


0^ 





9 


0. 


1 


8 


1- 


2 


7 


2. 


3 


6 


3. 


4. 


B 


A. 



P.P. 



TABLE VIIT.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
10° 11° 



/ 


Lg. Vers. 




1 

2 
3 
4 


8.18162 
.18306 
.18450 
.18594 
•18738 


5 
6 
7 
8 
9 


8-18881 
• 19024 
.19167 
•19309 
•19452 


10 

11 
12 
13 
14 


8.19594 
•19736 
•19878 
•20019 
•20160 


15 
16 
17 
18 
19 


8.20301 
•20442 
•20582 
•20723 
•20833 


30 

21 
22 
23 
24 


8^21003 
•21142 
•21282 
•21421 
•21560 


25 
26 
27 
28 
29 


8-21698 

21837 

.21975 

•22113 

•22251 


30 

31 
32 
33 
34 


8-22389 
•22526 
•22663 
•22800 
•22937 


35 
36 
37 
38 
39 


8-23073 
•23209 
•23346 
•23481 
•23617 


40 

41 
42 
43 
44 


8-23752 
.23«88 
•24023 
•24158 
•24292 


45 
46 
47 
48 
49 


8^24426 
•24561 
•24695 
-24828 
-24982 


50 

51 
52 
53 
54 


8-25095 
•25228 
-25361 
-25494 
-25627 


55 
56 
57 
58 
59 

60 


8-25759 
-25891 
•26023 
•26155 
•26286 

8.26417 


/ 


Lg.Vers. 



Losr.Exs. 



143 
143 
142 
142 
142 

14 

142 

142 

141 

141 

141 
140 
140 
140 
140 

140 
139 
139 
139 
139 

138 
138 
138 
138 
137 
138 
137 
137 
136 
137 

136 
138 
136 
135 
136 

135 
135 
135 
135 
134 

134 
134 
134 
133 
133 

133 
133 
133 
132 
133 
132 
132 
132 
132 
131 
131 



18827 
18973 
19120 
19266 
19411 



19557 
19702 
19847 
19992 
20137 



20281 
20425 
20589 
20713 
20857 



7> 



2100J 
21143 
21286 
21428 
21571 



21713 
21855 
21996 
22138 

22279 



22420 
22561 
22701 
22842 
22982 



23122 
23262 
23401 
23540 
23879 



23818 
23957 
24095 
24234 
24372 



24509 
24647 
24784 
24922 
25059 



25195 
25332 
25468 
25604 
25740 



25876 
26012 
26147 
26282 
26417 



26552 
26686 
26821 
26955 
27089, 



27228 



Log, Exs. 



14g 
146 
146 
145 

145 
145 
145 
145 
144 

144 
144 
144 
144 
143 

143 
143 
143 
142 
142 

142 
142 
141 
141 
141 

141 
140 
140 
140 
140 

140 
140 
139 
139 
139 

139 
138 
138 
138 
138 

137 
138 
137 
137 
137 

136 
136 
136 
136 
136 

136 
135 
135 
135 
135 

134 
134 
134 
134 
134 

134 



Lg. Vers. 



26417 
26548 
26679 
26810 
26941 



27071 
27201 
27331 
27461 
27590 



27719 
27849 
27977 
28106 
28235 



28363 
28491 
28619 
28747 
28875 



29002 
29129 
29256 
29383 
29510 



29636 
29763 
29889 
30015 
30140 



30266 
30391 
30516 
30642 
30766 



30891 
31015 
31140 
31264 
3138R 



31511 
31635 
31758 
31882 
32005 



32128 
32250 
32373 
32495 
32817 



32739 
32861 
32983 
33104 
33225 



33347 
33468 
33588 
33709 
33829 



33*^50 



» |Lg. Vers 



131 
131 
131 
130 

130 

130 
130 
130 
129 
129 
129 
128 
129 
128 

128 
128 
128 
128 
127 

127 
127 
127 
127 
126 

126 
126 
126 
126 
125 

125 
125 
25 
125 
124 

124 
124 
124 
124 
124 
123 
124 
123 
123 
123 

123 
122 
122 
122 
122 

122 
122 
121 
121 
121 

12l 
12] 
120 
120 
120 

120 

Id 



Log. Exs 



8 • 27223 
.27356 
.27490 
.27623 
•27756 



8 •27889 
•28021 
•28153 
•28286 
•28 418 

8.28550 
•28681 
•28813 

.28944 
.29075 



8.29206 
.29336 
.29467 
.29597 
•29727 

8.29857 
29987 
30117 
30246 
30375 



8.30504 
30633 
30762 
30890 
31019 



8.31147 
•31275 
•31402 
•31530 
•31657 



J> 



8^31785 
•31912 
.32039 
.32165 
.32292 



8.32418 
.32544 
.32670 
.32796 
.32922 



8-33047 
.33173 
.33298 
.33423 
^33547 

8.33672 
.33797 
.33921 
.34045 
^34169 

• 34293 
•34417 
-34540 
.34663 
.34786 



B. 34909 



Log. Exs 



133 
133 
133 
133 

133 
132 
132 
132 
132 

132 
131 
131 
131 
131 

131 
130 
130 
130 
130 

130 
130 
129 
129 
129 

129 
129 
128 
128 
128 

128 
128 

127 
127 
127 

127 
127 
127 
126 
126 

126 
126 
126 
126 
125 

125 
125 
125 
125 
124 

125 
124 
124 
124 
123 

124 
124 
123 
123 
123 

123 



P. P. 



10 

11 
12 
13 

14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
2± 
25 
28 
27 
28 
29 

30 

31 
82 
33 

34 

35 
36 
37 
38 



40 

41 
42 
43 
44 

45 
46 
47 
48 

li 

50 

51 
52 
53 
-54 

55 
56 
57 
58 
59 
60 





130 


12 


6 


13.0 


12. 


7 


15 


1 


14. 


8 


17 


3 


16. 


9 


19 


5 


18. 


10 


21 


6 


20. 


20 


43 


3 


40. 


30 


65 





60. 


40 


86 


6 


80- 


50 


108 


3 


100. 



6 

7 

8 

9 

10 

20 

30 

40 

50 






4 





4 


0. 





5 





4 


0. 





6 





5 


0. 





7 





6 


0. 





7 





6 


0^ 


1 


5 


1 


3 


1^ 


2 


2 


2 





I- 


3 





2 


6 


2. 


3 


7 


3 


3 


2. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 






3 


0. 





3 


0^ 





4 


0^ 





4 


0^ 





5 


0. 


1 





0- 


1 


5 


1- 


2 





1^ 


2 


5 


2. 






2 


0. 





2 


0. 





2 


0. 





3 


0. 





3 


0. 





6 


0. 


1 





0. 


1 


3 


1. 


1 


6 


1. 



1 




O 


0-1 


0- 





1 


0. 





1 


0. 





1 


0. 





1 


0. 





3 


0. 





5 


0- 





6 


0. 





8 


0. 



2 

2 
3 
3 

4 
4 
8 
2 
6 
1 



P. P. 



597 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
13° 13° 



O 

1 

2 

3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 
14^ 

15 
16 
17 
18 

IL 
30 

21 
22 
23 
2£ 

25 

26 

27 

28 

29_ 

IW 

31 

32 

33 

34^ 

35 

36 

37 

38 

39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49^ 

50 

51 
52 
53 
5£ 

55 

56 

57 

58 

^ 

60 



Lg. Vers 



33950 
34070 
34190 
34309 
34429 



34549 
34668 
34787 
34906 
35025 



35143 
35262 
35380 
35498 
35616 



35734 
35852 
35969 
36086 
36204 



36321 
36437 
36554 
36671 
36787 



36903 
37019 
37135 
37251 
37366 



37482 
37597 
37712 
37827 
37942 



38057 
38171 
38286 
38400 
■^8514 



38628 
38741 
38855 
38969 
39082 



39195 
39308 
39421 
39534 
39646 



39758 
39871 
39983 
40095 
40907 



40318 
40430 
40541 
40652 
40764 



«• 40875 



Lg. Vers. 



1> 

120 
120 
119 
120 

119 
119 
119 
119 
119 

118 
118 
118 
118 
118 

117 
118 
117 
117 
117 

117 
116 
117 
116 
116 

116 
116 
116 
115 
115 

115 
115 
115 
115 
115 

114 
114 
114 
114 
114 

114 
113 
114 
113 
113 

113 
113 
113 
113 
112 

112 
112 
112 
112 
112 

111 
111 
111 
111 
111 

111 



Log.Exs. 



I) 



34909 
35032 
35155 
35277 
35399 



35522 
35644 
35765 
35887 
36009 



36130 
36251 
36372 
36493 
36614 



36734 
36855 
36975 
37095 
37215 



37335 
37454 
37574 
37693 
37812 



37931 
3805C 
38169 
38287 
38406 

38524 
38642 
38760 
38878 
38995 

39113 
39230 
39347 
39484 
39'=^^T 



39698 
39814 
39931 
4004^ 
4016'^ 



4027^^ 
40395 
40511 
40626 
4074.? 



40857 
4097? 
41087 
41202 
41317 



41431 
4154B 
41660 
41774 
41888 
4?nn? 



123 
122 
122 
122 
122 
122 
121 
122 
121 

121 
121 
121 
120 
121 

120 
120 
120 
120 
120 

120 
119 
119 
119 
119 

119 
118 
119 
118 
118 

118 
118 
118 
118 
117 
117 
117 
117 
117 
117 

116 
116 
116 
116 
116 

116 
116 
115 
115 
115 

115 
115 
115 
115 
114 

114 
114 
114 
114 
114 

114 



Lg. Vers 



1> Log.Exs. I I> Lg. Vers 



40875 
40985 
41096 
41206 
41317 



41427 
41537 
41647 
41757 
41867 



41976 
42086 
42195 
42304 
42413 



42522 
42630 
42739 
42847 
42956 



43064 
43172 
43280 
43388 
43495 



43603 
43710 
43817 
43924 
44031 



44138 
44245 
44351 
44458 
44564 

44670 
44776 
44882 
44988 

45093 



45199 
45304 
45409 
4551? 
45619 



45724 
45829 
4593^ 
4603P 
46142 

46247 
46351 
4645F 
4655P 
46662 



4676P 

46862 
46972 
4707P 
47179 

4728? 



D 

llQ 

110 
110 
110 

110 
110 
110 
109 
110 

109 
109 
109 
109 
109 

109 
108 
109 
108 
108 
108 
108 
108 
108 
107 

107 
107 
107 
107 
107 
106 
107 
106 
106 
106 

106 
106 
105 
106 
105 

105 
105 
105 
105 
105 

105 
104 
105 
10£ 
104 

104 
104 
104 
103 
104 

105 
103 
103 
103 
103 
103 



Log. Exs 



42002 
42116 
42229 
42343 
42456 



42569 
42682 
42795 
42908 
43021 



43133 
43246 
43358 
43470 
43582 



43694 
43805 
43917 
44028 
44139 



44251 
44362 
44473 
44583 
44694 



44804 
44915 
45025 
45135 
45245 



4535b 
45465 
45574 
45684 
45793 



45902 
46011 
46120 
46229 
46338 



46446 
46555 
46663 
4677T 
46879 



46987 
47095 
47?0? 
47310 
47/117 



47525 
47632 
47739 
47846 
4795 3 
48060 
4816P 
48273 
48379 
48485 



48 591 



2> 

113 
113 
113 
113 

113 
113 
113 
113 
112 

112 
112 
112 
112 
112 

112 
ill 
111 
111 
111 
111 
HI 
111 
110 
110 

110 
110 

lie 
lie 

109 
110 

lie 

109 
109 
109 

109 
109 
lOG 
108 
109 

108 
108 
108 
108 
108 

loe 

107 

icr 

107 
107 

107 
107 
107 
107 
106 

107 
106 
lOP 
106 
lOP 

106 



1> Log.Exs., 
598 



D 





1 
2 
3 
4 

5 
6 
7 
8 
_9^ 

10 

11 
12 
13 
ii 
15 
16 
17 
18 
19 

30 

21 
22 
23 
2^ 

25 
26 
27 
28 

29 

30 

31 
32 
33 
34 

35 
36 
37 
38 

11 
40 

41 
42 
43 
44 

45 
46 
47 
48 

49 

50 

51 
52 
53 

54 



55 
56 
57 
58 
59 

BO 



P.P. 





130 


119 


6 


12.0 


11.91 


7 


14 





13 


9 


8 


16 





15 


8 


9 


18 





17 


8 


10 


20 





19 


8 


20 


40 





39 


6 


30 


60 





59 


5 


40 


80 





79 


3 


50 


100 





99 


1 





117 116 


11 


6 


11-7 


11.6 


11. 


7 


13 


6 


13 


5 


13. 


8 


15 


6 


15 


4 


15. 


9 


17 


5 


17 


4 


17. 


10 


19 


5 


19 


3 


19. 


20 


39 





38 


6 


38. 


30 


58 


5 


58 





57. 


40 


78 





77 


3 


76. 


50 


97 


5 


96 


6 


95. 



118 

11.8 
13-7 
15-7 
17.7 
19-6 
S9.3 
59.0 
78-6 
98.3 



114 113 



6 

7 

8 

9 

10 

20 

30 

40 

50 



11 


4 


11 


3 


11. 


13 


3 


13 


2 


13- 


15 


2 


15 





14. 


17 


1 


16 





16. 


19 





18 


8 


18. 


38 





37 


6 


37. 


57 





56 


5 


56- 


76 





75 


3 


74. 


95 





94 


1 


93. 



113 

2 

9 
8 
6 
3 

6 
3 





111 110 


109 


6 


11.1 


11.0 


10.9 


7 


12 


9 


12 


8 


12 


7 


8 


14 


8 


14 


6 


14 


5 


9 


16 


6 


16 


5 


16 


3 


10 


18 


5 


18 


3 


18 


1 


20 


37 





36 


6 


36 


3 \ 
5 


30 


55 


5 


55 





54 


40 


74 





73 


3 


72 


6 • 


50 


92 


5 


91 


6 


90 


8 



6 
7 
8 

9 
10 
20 
30 
40 
50 



108 

10. 8 

12 



107 



106 

10-6 
12 





105 


104 


0_ 


p 


10-5 


10.4 


0.0 


7 


12 


2 


12 


1 


0.0 


8 


14 





13 


8 


0.0 


q 


15 


7 


15 


6 


0.1 


10 


17 


5 


17 


3 


0.1 


?0 


35 





34 


6 


O.I 


30 


52 


5 


52 





0.2 


40 


70 





69 


3 


0.3 


50 


87 


5 


86 


6 


0.4 



P. P: 



i 



TABLEVIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
14° 15° 



Lg. Vers. -D Log.Exs. J> Lg. Vers. D 



47282 
47384 
47487 
47590 
47692 



47795 
47897 
47999 
48101 
48203 



48304 
48406 
48507 
48609 
48710 



48811 
48912 
49013 
49114 
4S215 



49315 
49415 
49516 
49616 
49716 



49816 
49916 
50015 
50115 
50215 



50314 
50413 
50512 
50611 
50710 



50809 
50908 
51006 
51105 
5120^ 



51301 
51399 
51497 
51595 
51693 



51791 
51888 
51986 
52083 
52180 



52277 
52374 
52471 
52568 
52665 



52761 
52858 
52954 
53050 
53146 



8.53242 



Lg. Vers 



102 
103 
102 
102 
102 
102 
102 
102 
102 

101 
101 
101 
101 
101 

101 
101 
101 
100 
101 

100 
100 
100 
100 
100 

100 

log 

99 
100 
99 
99 
99 
99 
99 
99 

98 
99 
98 
98 
98 
98 
98 
98 
98 
97 
98 
97 
97 
97 
97 
97 
97 
97 
96 
97 

96 
96 
96 
96 
96 



48591 
48697 
48803 
4890? 
49014 



49120 
49225 
49331 
49436 
49541 



49646 
49750 
49855 
49960 
50064 



50168 
50273 
50377 
50481 
50585 



50688 
50792 
50896 
50999 
51102 



51205 
51309 
51412 
51514 
51617 



51720 
51822 
51925 
52027 
52129 



52231 
52333 
52435 
52537 
52B3? 



5274C 
52841 
52943 
53044 
53145 



53245 
53347 
53448 
53548 
53649 



53749 
53850 
53950 
54050 
541 5r 



542 50 
5435C 
54449 
54549 
54649 



8-54748 
Log.Exs. 



106 
106 
105 
105 

105 
105 
105 
105 
105 

105 
104 
105 
104 
104 

104 
104 
104 
104 
104 

103 
104 
103 
103 
103 

103 
103 
103 
102 
103 

102 
102 
102 
102 
102 

102 
102 
102 
101 
101 

101 
101 
101 
101 
101 

101 
101 
101 
100 
100 

100 
100 
100 
100 
100 

100 

100 

99 

100 

99 

99 



8-53242 
53338 
53434 
53530 
53625 



53721 
53816 
53911 
54007 
54102 



54197 
54291 
54386 
54481 
54575 



54670 
54764 
548581 
54952 
55046 



55140 
55234 
55328 
55421 
5551 5 

55608 
55701 
55795 
55888 
55981 



56074 
56166 
56259 
56352 
56444 



56536 
56629 
56721 
56813 
56905 



56997 
57089 
57180 
57272 
57363 



57455 
57546 
57687 
5772^ 
57819 



5791C 
58001 
58092 
58182 
58273 



58363 
58453 
58544 
58634 
58724 

58814 



1> jLg. Vers. 



Log.Exs. JO 



8-54748 
54847 
54946 
55045 
55144 



55243 
55342 
55441 
55539 
55638 



55736 
55834 
55933 
56031 
56129 



56226 
56324 
56422 
56519 
56617 



56714 
56812 
56909 
57006 
57103 



57200 
57296 
57393 
57490 
57586 



57682 
57779 
57875 
57971 
58067 



58163 
58259 
58354 
58450 
58548 



58641 
58736 
58832 
58927 
59022 

5S117 
59211 
59306 
59401 
59495 



59590 
59684 
59779 
59873 
59907 



80061 
60155 
6024e 
60342 
80436 



8-60530 
Log.Exs. 



99 
99 
99 
99 

99 
9? 
98 
98 
98 

98 
98 
98 
98 
98 

97 
98 
97 
97 
97 
97 
97 
97 
97 
97 

97 
96 
97 
96 
96 

96 
96 
96 
96 
98 

95 
96 
95 
96 
95 

95 
95 
96 
95 
95 
9- 
94 
95 

9^ 
94 

94 
94 
94 
94 
94 

94 
94 
94 
93 
94 

93 

77 



5 

6 
7 
8 
9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 



20 
21 
22 
23 
24 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 

40 

41 
42 
43 
j44 

45 
46 
47 
48 

50 

51 
52 
53 

55 
56 
57 
58 
-59 
60 



P.P. 





103 


103 


6 


10.3 


10-21 


7 


12 





11 


9 


8 


13 


7 


13 


6 


9 


15 


4 


15 


3 


10 


17 


1 


17 





20 


34 


3 


34 





30 


51 


5 


51 





40 


68 


6 


68 





5C 


85 


b 


85 






101 

10.1 
11.8 
13.4 
15.1 
16-8 
33.6 
50-5 
67.3 
84.1 



100 

6 10-0 

7 11 

8 13 

9 15 
10 16 
20 33 
30;50 
40'66 
50183 



99 

9.9 



98 

9-8 
5 11-4 
2 13-0 
8 14-7 
5 16.3 
32.6 
549.0 
65.3 
5J82.6 





97 


96 


6 


9.7 


9-61 


7 


11 


3 


11 


2 


8 


12 


9 


12 


8 


9 


14 


5 


14 


4 


10 


16 


1 


16 





20 


32 


3 


32 





30 


48 


5 


48 


c 


40 


64 


6 


64 





50 


80 


8 


80 






95 

9.5 
11.1 
12.6 
14.2 
15.8 
31.6 
47.5 
63.3 
79.1 





94 


93 


9? 


6 


9 4 


9-3 


9. 


7 


10 


9 


10-8 


10. 


8 


12 


<j 


12.4 


12^ 


9 


14 


1 


13.9 


13- 


10 


15 


6 


15.5 


15. 


20 


31 


3 


31.0 


30. 


30 


47 





46.5 


4G. 


4G 


62 


6 


62.0 


01. 


50 


78 


3 


77.5 


76- 



91 

9-1 
10-6 
12.1 
13.6 
15.1 
30.3 
45.5 
60.6 
75.8 



90 

9-0 
10.5 
12. C 
13.5 
15. C 
30. C 
45.0 
60.0 
75. CIO. 4 



O 

0.0 
0.0 
0.0 
0.1 
0.1 
O.I 
0.2 
0.3 



P.P. 



599 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
16° 17° 





1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
14 
15 
16 
17 
18 

11 

20 

21 

22 

23 

24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59 

^0 



Lg.Vers. X> 



58814 
58904 
58993 
59083 
59J,73 

59262 
59351 
59441 
59530 
59619 



59708 
59797 
59886 
59974 
60063 



60152 
60240 
60328 
60417 
60505 



60593 
60681 
60769 
60857 
60944 

61032 
61119 
61207 
61294 
61381 



61469 
61556 
61643 
61730 
61816 



61903 
61990 
62076 
62163 
62249 



62336 
62422 
62508 
62594 
62680 



62766 
62852 
62937 
63023 
63108 



63194 
6327? 
63364 
6344? 
63534 



6361? 
63704 
63789 
63874 
63959 



8 64043 
Lg. Vers. 



90 
89 
90 
89 
89 
8? 
89 
89 
89 

89 
89 
89 
88 
89 

88 
88 
88 
88 
88 

88 
88 
88 
88 
87 

87 
87 
87 
87 
87 

87 
87 
87 
87 
86 

87 
86 
86 
86 
86 
86 
86 
86 
86 
86 
86 
86 
85 
85 
85 

85 
85 
85 
85 
85 

85 
85 
85 
84 
85 
84 



Log.Exs. I> 



60530 
60623 
60716 
60810 
60903 



60996 
61089 
61182 
61275 
61368 



61460 
61553 
61645 
61738 
6183C 



61922 
62014 
62106 
62198 
62290 

62382 
62474 
62565 
62657 
62748 



62840 
62931 
63022 
63113 
63204 



63295 
63386 
63477 
63567 
63658 



63748 
63839 
6392? 
6401? 
64109 



6419? 
64289 
64379 
64469 
64559 



6464? 
64738 
64828 
64917 
65006 



65096 
65185 
65274 
65363 
65452 



65541 
65629 
65718 
65807 
65895 
__ 65984 
1> Log.Exs. 



93 
93 
93 
93 

93 
93 
93 
92 
93 
92 
92 
92 
92 
92 

92 
92 
92 
92 
92 

91 
92 
91 
91 
91 

91 
91 
91 
91 
91 

90 
91 
91 
90 
90 

90 
90 
90 
90 
90 

90 
90 
90 
90 
89 
90 
89 
89 
89 
89 

89 
89 
89 
89 
89 

89 
88 
88 
8? 
88 

88 



Lg. Vers. 



64043 
64128 
64212 
64296 
64381 



64465 
64549 
64633 
64717 
64801 



64884 
64968 
65052 
65135 
65218 



65302 
65385 
65468 
65551 
65634 



65717 
65800 
65883 
65965 
66048 



66131 
66213 
66295 
66378 
66460 



66542 
66624 
66706 
66788 
66870 



66951 
67033 
67115 
67196 
67277 



6735? 
67440 
67521 
67602 
67683 



67764 
67845 
67926 
68007 
68087 



68168 
68248 
6832? 
6840? 
68489 



68569 
68650 
68730 
68810 
68889 



68969 



I> Lg.Vers. 



2> 



84 
84 
84 
84 

84 
84 
84 
84 
84 

83 
83 
84 
83 
83 

83 
83 
83 
83 
83 

83 
83 
82 
82 
83 

82 
82 
82 
82 
82 

82 
82 
82 
82 
82 

81 
81 
82 
81 
81 

81 
81 
81 
81 
81 

81 
81 
80 
81 
80 

80 
80 
80 
80 
80 

80 
80 
80 
80 
79 
80 



JJ 



Log.Exs. 



8.65984 
66072 
66160 
66248 
66336 



66425 
66512 
66600 
66688 
66776 



66863 
66951 
67039 
67126 
67213 



67301 
67388 
67475 
67562 
67649 



67736 
67822 
67909 
67996 
68082 



6816? 
68255 
68341 
68428 
68514 



6860C 
63686 
68772 
68858 
68944 



6902? 
69115 
69201 
69286 
69372 



69457 
69542 
69627 
69712 
69798 



69883 
69967 
70052 
70137 

70222 



70306 
70391 
70475 
70560 
70644 



70728 
70813 
70897 
70981 
71065 



8.71149 



Log.Exs, 



n 

88 
88 
88 
88 

88 
87 
88 
88 

87 

87 
88 
87 
87 
87 

87 
87 
87 
87 
87 

87 
86 
87 

86 
86 

86 
86 
86 
86 
86 

86 
86 
85 
86 
86 

85 

86 
85 
85 
85 

85 
85 
85 
85 
85 

85 
84 
85 
85 

84 

84 
84 
84 
84 

84 

84 
84 
84 
84 
84 

84 

77 





1 

2 
3 
4 

5 
6 
7 
8 
__9^ 

10 

11 
12 
13 
li 
15 
16 
17 
18 
ip 

20 
21 
22 
23 

,24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
M. 
35 
36 
37 
38 
89 

40 

41 
42 
43 



45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
li 
00 



P.P. 





93 


92 


6 


9.3 


9.2 


7 


10 


8 


10.7 


8 


12 


4 


12.2 


9 


13 


9 


13.8 


10 


15 


5 


15.3 


20 


31 





30.6 


30 


46 


5 


46.0 


40 


62 





61.3 


50 


77 


5 


76.6 





90 


89 


6 


9.0 


8.9 


7 


10 


5 


10.4 


8 


12 





11.8 


9 


13 


5 


13.3 


10 


15 


C 


14.8 


20 


30 


C 


29.6 


3C 


45 





44.51 


40 


60 


C 


59.3 


50 


75 





74.1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



87 
8.7 
10.1 
11.6 
13.0 
14.5 
29.0 
43.5 
58.0 



72.5 



86 

8.6 



91 

9.1 
10.6 
12.1 
13.6 
15.1 
30.5 
45.5 
60.6 
75.8 

88 
8.8 

10.2 
11.7 
13.2 
14.6 
29.3 
44.0 
58.6 
73.3 

85 
85 





84 


83 


82 


6 


84 


83 


8. 


7 


9 


8 


9.7 


9. 


8 


11 


2 


11-0 


10. 


9 


12 


6 


12.4 


12. 


10 


14 





13-8 


13. 


20 


28 





27.6 


27. 


30 


42 





41.5 


41. 


40 


56 





55.3 


54. 


50 


70 





69.1 


68. 





81 


80 


6 


8.1 


8.01 


7 


9.4 


9 


3 


8 


10.8 


10 


6 


9 


12.1 


12 





10 


13.5 


13 


3 


20 


27.0 


26 


6 


30 


40.5 


40 





40 


54.0 


53 


3 


50 


67.5 


66 


6 



79 

79 
9.2 
10.5 
11.8 
13.1 
26.3 
39.5 
52.6 
65-8 



6 
7 
8 

9 
10 
20 
30 
40 

FT 



o 

0.0 
0.0 
0.0 
0.1 
0.1 
O.I 
0.2 
O-S 



P.P. 



600 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 
18° 19° 



Lg. Vers, 



8.68969 
69049 
69129 
69208 
69288 



55 
56 
57 
58 
59 
60 



69367 
69446 
69526 
69605 
69684 



69763 
69842 
69921 
70000 
70079 



70157 
70236 
70314 
70393 
7047T 



70050 
70628 
70706 
70784 
70862 



70940 
71018 
71096 
71174 
71251 



71329 
71406 
71484 
71561 
71639 



71718 
71793 
71870 
71947 
720^4 



D Log.Exs. 



72101 
72178 
72255 
72331 
72403 



72485 
72561 
72637 
72714 
72790 



72886 
72942 
73018 
73094 
73170 



73246 
73322 
73398 
7347? 
73549 



R .7r5^9Pi 



79 
80 
79 
79 

79 
79 
79 
79 
79 

79 
79 
79 
78 
79 

78 
78 
78 
78 
78 

78 
78 
78 
78 
78 

78 
78 

77 
78 

77 

77 
77 
77 
77 
77 

77 
77 
77 
77 
77 

77 
76 
77 
76 
77 

76 
76 
76 
76 
76 

76 
76 
76 
76 
76 

76 
76 
75 
75 
76 

75 



8-71149 
.71232 
.71316 
.71400 
•71484 



8.71567 
.71651 
.71734 
.71817 
.71901 



8-71984 
.72067 
-72150 
-72233 
•72316 



8-72399 
-72481 
-72564 
-72847 
-72729 



8-72812 
-72894 
-72977 
•73059 
-73141 



8.7322b 
-73306 
-73388 
-73470 
•73551 



8-73833 
-73715 
.73797 
.73878 
-73960 



8 - 74041 
-74123 
-74204 
-74286 



Lg. Vers, 



8 • 74448 
-74529 
-74610 
-74691 
•74772 



8-74853 
•74934 
.75014 
.75095 
•75175 



8^75256 
•75336 
.75417 
•75497 
•75577 



I) 



8-75658 
-75738 
-75818 
-75898 
•75978 



8.7RnFi8 



Log.Exs, 



83 
84 
83 
84 

83 
83 
83 
83 
83 

83 
83 
83 
83 
83 
83 
82 
83 
82 
82 

82 
82 
82 
82 
82 

82 
82 
82 
82 
81 

82 
82 
81 
81 
81 

8l 
81 
81 
81 
81 

81 
81 
81 
81 
80 

81 
81 
80 
80 
80 

80 
80 
80 
80 
80 

80 
80 
80 
80 
80 

80 



Lg.Vers, 



8.73625 
.73700 
.73775 
.73851 
-73926 



8.74001 
.74076 
.74151 
.74226 
•74301 



8-74376 
-74451 
.74526 
-74600 
•74675 



8 - 74749 
-74824 
.74898 
-74973 
-75047 



8-75121 
.75195 
•75269 
.75343 
-75417 



8-75491 
•75565 
•75639 
.75712 
-75786 



8-75860 
•75933 
.76006 
.76080 
-76153 



8.76226 
.76300 
.76373 
.76446 
-76519 



8.76592 
.76664 
.76737 
•76810 
-76883 



8-76955 
•77028 
.77100 
.77173 
-77245 



8-77317 
•77390 
•77462 
•77534 
-77606 



8.77678 
•77750 
•77822 
•77893 
-77965 



8.78037 



75 
75 
75 
75 
75 
75 
75 
75 
75 

75 
74 
75 
74 
74 

74 
74 
74 
74 
74 
74 
74 
74 
74 
74 

74 
73 
74 
73 
73 

74 
73 
73 
73 
73 

73 
73 
73 
73 
73 

73 
72 
73 
72 
73 

72 
72 
72 
72 
72 

72 
72 
72 
72 
72 

72 
72 
72 
71 
.72 
71 



Lg. V^rs, 



Log.Exs, 



76058 
76137 
76217 
76297 
76376 



76456 
76536 
76615 
76694 
76774 



76853 
76932 
77011 
77090 
77169 



77248 
77327 
77406 
77485 
77563 



77642 
77720 
77799 
77877 
77956 



78034 
78112 
78191 
78269 
78347 



78425 
78503 
78581 
78659 
78736 



78814 
78892 
78969 
79047 
79124 



79202 
79279 
79357 
79434 
79511 



79588 
79665 
79742 
79819 
79896 



79973 
80050 
80126 
80203 
80280 



80356 
80433 
80509 
80586 
80662 



80738 



79 
80 
79 
79 
80 
79 
7? 
79 
79 

79 
79 
79 
79 
79 

79 
79 
78 
79 
78 

78 
78 
79 
78 
78 

78 
78 
78 
78 
78 

78 
78 
78 
78 

77 

78 

77 
77 
77 
77 

77 
77 
77 
77 
77 

77 
77 
77 
77 
77 
76 
77 
76 
77 
76 

76 
76 
76 
76 
76 
76 



J> Log.Exs.i2> 
~601 



10 

11 

12 
13 
li 
15 
16 
17 
18 
19 

SO 

21 
22 
23 
24 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 



35 
36 
37 
38 
39, 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 

54 

55 
56 
57 
58 
_59 
60 



P.P. 





84 


83 


8? 


6 


8.4 


8-3 


8. 


7 


9-8 


9-7 


9^ 


8 


11-2 


11-0 


10. 


9 


12-6 


12-4 


12. 


10 


14-0 


13-8 


13. 


20 


28-0 


27^6 


27^ 


30 


42-0 


41-5 


41. 


40 


56-0 


55.3 


54- 


50 


70-0 


69.1 


68- 



81 80 



8 


1] 


8-0 


7- 


9 


4 


9-3 


9. 


10 


8 


10-6 


10^ 


12 


1 


12^0 


11. 


13 


5 


13-3 


13. 


27 





26-6 


28. 


40 


5 


40-0 


39. 


54 





53 • 3 


52. 


67 


5 


66-6 


85. 





78 


77 


7( 


6 


7-8 


7.7 


7. 


7 


9 


1 


9 





8. 


8 


10 


4 


10 


2 


10. 


9 


11 


7 


11 


5 


11. 


10 


13 





12 


8 


12. 


20 


26 





25 


6 


25- 


30 


39 





38 


5 


38. 


40 


52 





51 


3 


50. 


50 


65 





64 


1 


63. 



75 


74 


7( 


7-5 


7-4 


7. 


8-7 


8^6 


8. 


10-0 


9.8 


9^ 


11-2 


11.1 


10^ 


12-5 


12.3 


12 • 


25-0 


24^6 


24 • 


37-5 


37^0 


36. 


50.0 


49^3 


48 • 


62-5 


61.6 


60- 





72 


"^1 . 


e 


7-2 


7.1| 


7 


8^4 


8 


3 


8 


9.6 


9 


A. 


9 


10^8 


10 


6 


10 


12.0 


11 


8 


20 


24.0 


23 


6 


30 


36.0 


35 


5 


40 


48^0 


47 


3 


50 


60.0 


59 


li 



79 

9 
2 
5 
8 
1 
3 
5 
6 
3 



O 

0.0 
0-0 
0.0 
0.1 
0.1 
01 
0.2 
0.3 
0.4 



P. P. 



TABLE VIIT.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
20° 31° 



Lg. Vers 



8.78037 
78108 
78180 
78251 
78323 



78394 
78466 
78537 
78608 
78679 



78750 
78821 
78892 
78963 
79034 



79105 
79175 
79246 
79317 
79387 



79458 
79528 
79598 
79669 
79739 



79809 
79879 
79949 
80019 
80089 



80159 
80229 
80299 
80369 
80438 



80508 
80577 
80647 
80716 
80786 



80855 
80924 

1 nc'> 



81058 
81132 



81201 
81270 
81339 
81407 
81476 



81545 
31614 
81682 
81751 
81819 



8188B 
81956 
82025 
82093 
8216] 



60? « • R2229 
*^|Lf^. Vers. 



n 



71 
71 
71 
71 

7l 
71 
71 
71 
71 

71 
71 
71 
71 
70 

71 
70 
71 
70 
70 

70 
70 
70 
70 
70 

70 
70 
70 
70 
70 

70 
70 
69 
70 
69 

69 
69 
69 
69 
69 

69 
69 
69 

69 
69 

69 
69 
69 
68 
69 

68 
69 

68 
68 
68 

6£ 
6P 
6B 
6R 
68 
68 



Log.Exs. 



80738 
80814 
80891 
80967 
81043 



81119 
81195 
81271 
81346 
81422 



81498 
81573 
81649 
81725 
81800 



81876 
81951 
82026 
82102 
82177 

82252 
82327 
82402 
82477 
82552 



82627 
82702 
82776 
82851 
82926 

83000 
83075 
83149 
83224 
83298 



83373 

.83447 
83521 
83595 
83670 

83744 
83818 

.83S92 
83366 
84039 



8.84113 
.84187 

.84261 
•8433? 
.84408 

8.84481 

.84555 

.84628 

.84702 

84775 



D 



8 . 84848 

.84922 

.84995 

.85068 

85141 



3-35214 



Loff.Exs. 



76 
76 
76 
76 

76 
76 
76 
75 
76 

75 
75 
76 
75 
75 

75 
75 
75 
75 
75 

75 
75 
75 

75 
74 

75 
75 
74 
75 
74 

74 
74 
74 
74 
74 

74 
74 
74 
74 

74 

74 
74 

74 
74 
73 

74 
73 
74 
73 
73 

73 

73 
73 
73 
73 

73 
73 
73 
73 
73 

73 

"IF 



Lg. Vers. l>[Log.Exs. 



8.82229 
82297 
82366 
82434 
82502 



82569 
82637 
82705 
82773 
82841 



82908 
82976 
83043 
83111 
83178 



83246 
83313 
83886 
83447 
83515 



83582 
83649 
83716 
83783 
83850 

83916 
83983 
84050 
84117 
84183 



84250 
84316 
84383 
84449 
84515 



84582 
84648 
84714 
84780 

r4846 



84912 
84978 
35044 
8511D 
8517g 



85242 
85308 
85373 
85439 
85505 



855^0 
85626 
85*701 
85'766 
85832 

85897 
85962 
86027 
86092 
86158 



86223 



lg. Vers, 



68 
68 
68 
68 

67 
68 
68 
67 
68 

67 
67 
67 
67 
67 

67 
67 
67 
67 
67 

67 
67 
67 
67 
67 

66 
67 
66 
67 
66 

66 
66 
66 
66 
66 

66 
66 
66 
66 
66 

66 
66 

66 
€6 

66 

65 
66 
65 
65 
66 

65 
65 
65 
65 
65 

65 
65 
65 
65 
65 

65 
■^ 



85214 
85287 
85360 
85433 
85506 



85579 
85651 
85724 
85797 
85869 



85942 
86014 
86087 
86159 
86231 



86304 
86376 
86448 
86520 
86592 



86664 
86736 
86808 



86952 



87024 
87095 
87167 
87239 
87310 



87382 
87453 
87525 
87596 
87668 



87739 
87810 
87881 
87953 
88024 



88095 
88166 
88237 
88308 
88378 



88449 
88520 
88591 
88661 
88732 



88O03 
888'73 
88944 
89014 
89085 



89155 
89225 
89295 



89436 



89506 



Lcg.Exs. 



73 
73 
72 
73 

73 
72 
73 
72 
72 

72 
72 
72 
72 
72 

72 
72 
72 
72 
72 

72 
72 
72 
72 
71 
72 
7l 
72 
71 
71 

71 
71 
71 
71 
71 

71 
71 
71 
71 
71 
71 
71 
71 
71 
70 

71 
71 
70 
70 
71 

70 
70 
70 
70 
70 

70 
70 
70 
70 
70 

70 





1 

2 

3 

__£ 

5 

6 

7 

8 

_i. 

10 

11 

12 

13 

ii 

15 

16 

17 

18 

-ii 
20 

21 
22 
23 
2^ 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 



I> 



50 

51 
52 
53 

55 
56 
57 
58 
59 

60 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

2C 

30 

40 

50 



76 


75 


7.6 


7.5 


8.8 


8.7 


10.1 


10. C 


11.4 


11.2 


12.6 


12.5 


25.3 


25.0 


38. C 


37.5 


50.6 


50. G 


63.3 


62.5 



74 

7-4 

8.6 

9.8 

11.1 

12.3 

21.6 

37.0 

49.8 

61.6 



73 

7.3 

5 
7 
9 

i 

3 

6 
8 



72 
7.2 



71 

7.1 
4| 8.3 
6' 9.4 
810.6 
11.8 
23.6 
035.(3 
047-3 

o;59.i 



70 

7-0! 



69 

6-9 



1! 8 
3 9 
510 
6 11 
3 23 
34 
6 46 
357 



68 

6 8 

7 9 
9-0 





67 


66 


6 


6.7 


6.6i 


7 


7 


8 


7 


7 


8 


8 


q 


8 


8 


9 


10 





9 


9 


10 


11 


1 


11 





20 


22 


3 


22 


( 


30 


33 


5 


33 


C 


40 


44 


6 


44 





50 


55 


8 


55 






10 2 

11 3 
22- 6 
5:34 
0145.3 
5l5b.6 



65 

6.5 
7.6 
8.6 
9.7 
10.8 
21.6 



32. 
43.: 

54.: 



6 

7 

8 

9 

10 

20 

30 

40 

50 



O 

0.0 

0.0 

0.0 
0.1 
0.1 
O.I 
0.2 
0.3 
0.4 



P. P. 



602 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
22° 23° 



Lg.Vers.(2> Log.Exs. ■!> Lg.Vers. 1> Log.Exs. J> 



O 8.86223 



1 

2 

3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 
11 
15 
16 
17 
18 
19 



.86287 
.86352 
.86417 
•86482 



8.86547 
.86612 
.86678 
.86741 
.86805 



8.86870 
.86934 
.86999 
.87063 
.87127 



8.87192 
•87256 
.87320 
.87384 
•87448 



20 

21 
22 
23 
24 

25 

26 

27 

28 

2i 

30 

31 

32 

33 

34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



8-87512 
87576 
87640 
87704 
87788 



8.87832 
.87895 
•87959 
.88023 
•88086 



8-88150 
•88213 
•88277 
•88340 
•88404 



8 •88467 
•88530 
•88593 
•88656 
•88720 

8. 88783 

•88848 

•88909 

88971 

•89034 



8^C9097 
.89160 
.89223 
.89285 
•89348 



8-89411 
•89473 
•89536 
•89598 
•89680 



8^89723 
•89785 
•89847 
•89910 
•89972 



8 00034 
Lg, Vers. 



64 
65 
65 
65 

64 
65 
64 
64 
64 

64 
64 
64 
64 
64 

64 
64 
64 
64 
64 

64 
64 
64 
64 
63 

64 
63 
64 
63 
63 

63 
63 
63 
63 
63 

63 
63 
63 
63 
63 
63 
63 
63 
62 
63 

63 
62 
63 
62 
82 

63 
62 
62 
62 
62 

62 
62 
62 
62 
62 

82 



8^89b06 
.89576 
.89646 
.89716 
•89786 



8-89856 
•89925 
•89995 
.90065 
.90135 



8.90205 
.90274 
.90344 
.90413 
.90433 



8.90552 
.90822 
.90691 
90760 
90830 



8.90899 
•90968 
•91037 
.91106 
•91175 



8-91244 
.91313 
.91382 
•91451 
.91520 



8.9158S 
.91657 
.91726 
•91794 
.91863 



8.91932 
.92000 
.92068 
.92137 
•92205 

8-92274 
•92342 
.92410 
.92478 
.92546 



2> 



8-92615 
-92683 
-92751 
.92819 
.92887 



8.92955 
.93022 
-93090 
-93158 
•93226 



8^93293 
•93361 
•93429 
•93496 
•93564 



8. 93831 



xs, 



Log.E 



70 
70 
70 
69 

70 
70 
69 
70 
69 
70 
69 
69 
69 
69 

69 
69 
69 
69 
69 
69 
69 
69 
69 
69 

69 
69 
68 
69 
69 

68 
69 
68 
68 
68 

69 
68 
68 
68 
88 

68 
68 
68 
68 
68 

68 
68 
68 
68 
68 

68 
67 
68 
67 
68 

67 
68 
67 
67 
67 

67 



8. 



90034 
90096 
90158 
90220 
90282 



90344 
90406 
90467 
90529 
90591 



90652 
90714 
90776 
90837 
90899 



90980 
91021 
91083 
91144 
91205 



91267 
91328 
91389 
91450 
91511 



91572 
91633 
91694 
91755 
91815 



.91876 
.91937 
.91997 
-92058 
.92119 

.92179 
.92240 
.92300 
.92361 
-92421 

•92487 
•92542 
•92602 
•92862 
-92722 



8-92782 
-92842 
-92902 
-92962 
-93022 



8-93082 
•93142 
•93202 
•93261 
•93321 



8 •93381 
•93440 
-93500 
•93560 
•93619 



8-93879 
Lg» Vers 



62 
62 
62 
62 

62 
62 
61 
62 
61 

6l 
62 
61 
61 
61 

61 
61 
61 
61 
61 

61 
61 
61 
61 
61 

61 
61 
61 
61 
60 

61 
60 
60 
61 
60 

60 
60 
60 
60 
60 

60 
60 
60 
80 
60 
60 
60 
60 
60 
60 

60 
59 
60 
59 
60 

59 
59 
60 
59 
59 

59 



8-93631 
.93699 
.93766 
.93833 
.93901 



8.93968 
.94035 
.94102 
.94170 

.94237 



8.94304 
.94371 
.94438 
.94505 
•94572 



8 •94838 
.94705 
.94772 
.94839 
•94905 



•94972 
.95039 
•95105 
.95172 
.95238 



8.95305 
.95371 
.95437 
.95504 
•95570 



8.95636 
.95703 
.95769 
.95835 
.95901 



8.95987 
.96033 
•96099 
.96165 
•96231 



8 •96297 
•96362 
.96428 
.96494 
•96560 



8 •96825 
•96691 
•96757 
•96822 
•96888 



8^96953 
•97018 
•97084 
.97149 
.97214 



8.97280 
.97345 
.97410 
.97475 
.97540 



8. 97606 



Log.Exs. 



10 

11 
12 
13 
14 



15 
16 
17 
18 
11 
20 
21 
22 
23 
24 



P.P. 



30 

31 
32 
33 
34 



35 
38 
37 
38 

40 

41 
42 
43 
44 



50 

51 
52 
53 
54 



60 





70 


69 


6^ 


6 


7-0 


6.9 


6. 


7 


8.1 


8.0 


7. 


8 


9.3 


9.2 


9. 


9 


10.5 


10-3 


10. 


10 


11.6 


11.5 


11. 


20 


23-3 


23.0 


22. 


30 


35.0 


34.5 


34. 


40 


46.6 


46.0 


45- 


50 


58.3 


57.5 


56. 





67 


66 


6^ 


6 


6.7 


6.6 


6. 


7 


7 


8 


7 


7 


7. 


8 


8 


9 


8 


8 


8- 


9 


10 





9 


9 


9- 


10 


11 


1 


11 





10- 


20 


22 


3 


22 





21^ 


30 


33 


5 


33 





32^ 


40 


44 


6 


44 


C 


43 • 


50 


55 


8 


55 





54- 





64 


63 


65 


6 8-4 


6-3 


6- 


7 7 


4 


7 


3 


7- 


8 8 


5 


8 


4 


8- 


9 9 


6 


9 


4 


9. 


10 10 


6 


10 


5 


10- 


20 


21 


3 


21 


C 


20- 


30 


32 





31 


5 


31- 


40 


42 


6 


42 





41. 


50 


53 


3 


l52 


5 


51. 





61 


60 


59 


6 


6.1 


6-0 


5-9 


7 


7 


1 


7 





6 


9 


8 


8 


1 


8 





7 


8 


9 


9 


1 


9 





8 


8 


10 


10 


1 


10 





9 


8 


20 


20 


3 


20 





19 


6 


30 


30 


5 


30 





29 


5 


40 


40 


6 


40 





39 


3 


50 


50 


8 


50 





49 


1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



0% 

n 

0^1 
0.1 
0.1 
0.2 
0.3 
0.4 



P.P. 



603 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

34° 25° 



O 

1 
2 
3 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 

11 
30 
21 
22 
23 
24 

25 
26 
27 
28 
29 

ao 

31 
32 
33 

34 

35 
30 
37 
30 

po 

40 
41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Lg. Vers. 



93679 
93738 
93797 
93857 
93916 



93975 
94034 
94094 
94153 
94212 



94271 
94330 
94389 
94448 
9450'6 



94505 
94624 
94683 
94742 
94800 



94859 
94917 
94970 
95034 
95093 



95151 
95210 
95268 
95326 
95384 



95443 
95501 
95559 
95617 
95675 



95733 
95791 
95849 
95907 
nn905 



96023 
96080 
96138 
96196 
96253 



96311 
96368 
96426 
96483 
96541 

96598 
9665G 
96713 
96770 
96827 



96885 
96942 
96999 
97056 
97113 



8 97170 
Lg. Vers 



n 

59 
59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
58 

59 
59 
58 
59 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
57 
58 
58 

58 

57 
57 
58 
57 

57 
57 
57 
57 
57 

57 
57 
57 
57 
57 

57 
57 
57 
57 
57 

57 



Log.Exs. ^ 



8-97606 
97671 
97736 
97801 
97865 



97930 
97995 
98060 
98125 
9819C 



98254 
98319 
98383 
98448 
98513 



98577 
98642 
98706 
98770 
98835 



98899 
98963 
99028 
99092 
99156 



99220 
99284 
99348 
99412 
99476 



99540 
99604 
99668 
99732 
99796 



99860 
99923 
99987 
00051 
00114 



00178 
00242 
00305 
00369 
00432 



00495 
00559 
00622 
00686 
00749 

0081 1 
00875 
00938 
01002 
01065 



01128 
01191 
01254 
01317 
01380 



9.01443 
Log, Exs 



65 
65 
65 
64 

65 
65 
64 
65 
65 

64 
64 
64 
65 
64 

44 
64 
64 
64 
64 

64 
64 
64 
64 
64 
64 
64 
64 
64 
64 

64 
64 
64 
64 
63 

64 
63 
64 
63 
63 
64 
63 
63 
63 
63 

63 
63 

63 
63 
63 

63 

e3 

63 
63 
63 

63 
63 
63 
63 
63 
63 



Lg.Vers. ^ 



97170 
97227 
97284 
97341 
97398 



97455 
97511 
97568 
97625 
i7681 

97738 
97795 
97851 
97908 
97964 



98020 
98077 
98133 
98190 

98246 



98302 
98358 
98414 
98470 
98527 



98583 
98639 
98695 
98750 
98806 
988C2 
98918 
98974 
99030 
99085 



99141 
99197 
99252 
99308 
99363 



99419 
99474 
99529 
99585 
99640 



99695 
99751 
99806 
99861 
99916 

99971 
0002e 
00081 
00136 
00191 



00246 
00301 
00356 
00411 
00466 
00520 
Lg. Vers. 



57 
56 
57 
57 

57 
56 
57 
56 
56 
56 
57 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
5o 
56 

56 
56 
55 
56 
55 

55 
56 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55- 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
54 
55 

54 



Log.Exs, 



01443 
01505 
01568 
01631 
01694 



01756 
01819 
01882 
01944 
02GC7 



0207C 
02132 
02195 
02257 
02319 



C238z 
02444 
02506 
0256S 
02631 



C2G9 

02755 

02817 

0288C 

02942 

03004 
03066 
03128 
0319G 

03252 



03313 
03375 
03437 
03499 

03561 



03622 
03684 
03746 
03807 
038P9 



G39SC 
03992 
04053 
04115 
04176 



04238 
04299 
04360 
04421 
04483 



04544 
04605 
04666 
04727 
04788 



04850 
04911 
04972 
05033 
05093 



05154 
Log.Exs. 



n 

62 
63 
62 
63 
62 
63 
62 
62 
63 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

61 
62 
62 
61 
62 

61 
61 
62 
61 
61 

61 
61 
61 
61 
61 

6l 
61 
61 
61 
61 

61 
61 
61 
61 
61 

6l 
61 
61 
61 
60 

61 

17 



10 

11 

12 
13 
14 

15 
16 
17 
18 

Jl 
20 

21 
22 
23 
_24 
25 
26 
27 
28 

30 

31 
32 
33 
14 
35 
36 
37 
38 

40 

41 
42 
43 

M 
45 
46 
47 
48 

_49 

50 
51 
52 
53 

_54 

55" 
56 
57 
58 
59 
f?0 



P. P. 





65 


64 


6 


6.5 


6.4i 


7 


7.6 


7 


4 


8 


8.6 


8 


5 


9 


9.7 


9 


6 


10 


10.8 


10 


6 


20 


21.6 


21 


3 


30 


32.5 


32 





40 


43.3 


42 


6 


50 


54.1 


53 


3 



63 

6.3 

7.3 

8.4 

9.4 

10.5 

21.0 

31-5 

42.0 

52.5 



6 

7 
8 
9 
10 
20 
30 
40 
50 



62 61 60 



6 


2 


6 


1 


6. 


7 


2 


7 


1 


7. 


8 


2 


8 




8. 


9 


3 


9 




9. 


10 


3 


10 




10. 


20 


6 


20 


3 


20. 


31 





30 


5 


30. 


41 


3 


40 


5 


40. 


51 


6 


50 


8 


50. 





59 


58 


6 


5.9 


5. 81 


7 


6 


9 


6 


7 


8 


7 


8 


7 


7 


9 


8 


8 


8 


7 


10 


9 


8 


9 


c 


20 


19 


6 


19 


3 


3C 


29 


5 


29 


c 


40 


39 


3 


38 


6 


50 


49 


1 


48 


3 





56 


55 


6 


5.6 


5.5 


7 


6.5 


6.4 


8 


7.4 


7.3 


9 


8.4 


8-2 


10 


9.3 


9.1 


20 


18.6 


18.3 


30 


28.0 


27.5 


40 


37.3 


36.6 


50 


46.6 


45.8 



57 

5.7 

6.6 

7-6 

8.5 

9.5 

19.0 

28.5 

38.0 

47.5 



54 

5.4 

6.3 

7.2 

8.1 

9.0 

18.0 

27.0 

36.0 

45.0 



O 

0.0 
0.0 
0.0 
0.1 
0.1 
0.1 
0.2 
0.3 
0.4 



K. P. 



604 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
26° 27° 



Lg. Vers. 



00520 
00575 
00630 
00684 
00739 



00794 
00848 
00903 
00957 
01011 



01066 
01120 
01174 
01229 
01283 



01337 
01391 
01445 
01499 
01554 



01608 
01662 
01715 
01769 
01823 



01877 
01931 
01985 
02038 
02092 



02146 
02199 
02253 
02307 
02360 



02414 
02467 
02521 
02574 
02627 



02681 
02734 
02787 
02840 
02894 



02947 
03000 
03053 
03106 
03159 



03212 
03265 
03318 
03371 
03423 



03476 
03529 
03582 
03634 
03687 



9 03740 
Lg. Vers. 



2> Log.Exs 



55 
54 
54 
54 

55 
54 
54 
54 
54 
54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
53 
54 
54 

54 
53 
54 
53 
54 

53 
53 

54 
53 
53 
53 
53 
53 
53 
53 

53 
53 i 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
52 

53 
53 
52 
52 
53 
52 



05154 
05215 
05276 
05337 
05398 



05458 
05519 
05580 
05640 
05701 



05762 
05822 
05883 
05943 
06004 



06064 
06124 
06185 
06245 
06305 



06366 
06426 
06486 
06546 
06606 



06667 
06727 
06787 
06847 
06907 



06967 
07027 
07087 
07146 
07206 



07266 
07326 
07386 
07445 
07505 



07565 
07624 
07684 
07743 
07803 



07863 
07922 
07981 
08041 
08100 



08160 
08219 
08278 
08338 
08397 



08456 
08515 
08574 
08634 
08693 



08752 
Log.Exs 



jD 



61 
61 
60 
61 

60 
61 
60 
60 
60 

61 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
80 
60 
59 
60 

60 
59 
60 
59 
60 

59 
59 
59 
59 
60 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 



Lg. Vers, 



03740 
03792 
03845 
03898 
03950 



04002 
04055 
04107 
04160 
04212 



04264 
04317 
04369 
04421 
04473 



04525 
04577 
04630 
04682 
04734 



04786 
04837 
04889 
04941 
04993 



D 



05045 
05097 
05148 
05200 
05252 



05303 
05355 
05407 
05458 
05510 



05561 
05613 
05664 
05715 
05767 



05818 
05869 
05921 
05972 
06023 



06074 
06125 
06176 
06227 
06279 



06330 
06380 
06431 
06482 
06533 



06584 
06635 
06686 
06736 
06787 
06838 
Lg. Vers. 



52 
52 
53 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
51 
52 
52 
52 

5l 
52 
51 
52 
52 

5l 
52 
51 
51 
51 

51 
51 
5l 
51 
5l 

5l 
51 
5l 
51 
5l 

51 
51 
51 
51 
5l 

51 
50 
51 
51 
51 

51 
50 
51 
50 
51 

50 



Log.Exs, 



08752 
08811 
08870 
08929 
08988 

09047 
09106 
09164 
09223 
09282 



09341 
09400 
09458 
09517 
09576 



09634 
09693 
09752 
09810 
09869 



09927 
09986 
10044 
10102 
10161 



10219 
10278 
10336 
10394 
10452 



10511 
10569 
10627 
10685 
10743 



10801 
10859 
10917 
10975 
11033 



11091 
11149 
11207 
11265 
11323 



11380 
11438 
11496 
11554 
11611 



11669 
11727 
11784 
11842 
11899 



11957 
12015 
12072 
12129 
12187 



9-12244 



Log.Exs, 





1 

2 
3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
L4 

15 
16 
17 
18 
19 

20 

21 
22 
23 
2^ 

25 
26 
27 
28 
-29 

30 

31 
32 
33 
ii 
35 
36 
37 
38 

31 
40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



P. P. 





61 


60 


51 


6 


6.1 


6.0 


5. 


7 


7-1 


7 





6- 


8 


8.1 


8 





7- 


9 


9.1 


9 





8. 


10 


10.1 


10 





9. 


20 


20.3 


20 





19. 


30 


30.5 


30 





29. 


40 


40.6 


40 





39. 


50 


50.8 


50 





49. 





58 


57 


6 


58 


5.7 


7 


6 


7 


6 


6 


8 


7 


7 


7 


6 


9 


8 


7 


8 


5 


10 


9 


6 


9 


5 


20 


19 


3 


19 





30 


29 





28 


5 


40 


38 


6:38 





50 


48 


3i47 


5 



55 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



5 


5 


5 


6 


4 


6 


7 


3 


7. 


8 


2 


8- 


9 


1 


9. 


18 


3 


18. 


27 


5 


27- 


36 


6 


36. 


45 


8 


45. 



53 



5 


3 


5. 


6 


2 


6. 


7 





6 


7 


9 


7 


8 


8 


8 


17 


6 


17 


26 


5 


26 


35 


3 


34 


44 


1 


43 



'51. 


5.1! 


5 


9 


6 


8 


7 


6 


8 


5 


17 





25 


8 


34 





42 


5 



P.P. 



54 

4 
3 
2 
1 








52 
2 



O 

0.0 



G05 



TABLE VIII— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
28° 39° 



O 

1 

2 

3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 
14 
15 
16 
17 
18 

11, 
30 

21 
22 
23 
24 

25 
26 
27 
28 
2i 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

60 

51 
52 
53 
54 

55 
56 
57 
58 
51 
60 



Lg. Vers 



06838 
06888 
06939 
06990 
07040 



07091 
07141 
07192 
07242 
07293 



07343 
97393 
07444 
07494 
07544 



07594 
07644 
07695 
07745 
07795 



07845 
07895 
07945 
07995 
08045 

08095 
08145 
08195 
08244 
08294 



08344 
08394 
08443 
08493 
08543 



08592 
08642 
08691 
08741 
08790 



08840 
08889 
08939 
08988 
09087 



09087 
09136 
09185 
09234 
09284 



09333 
09382 
09431 
09480 
09529 



0957P 
09627 
09676 
09725 
09774 



9 09«9r 



Lg. Vers 



D 



n 



Log.Exs. J> 



12244 
12302 
12359 
12416 
12474 



12531 
12588 
12845 
12703 
12760 



12817 
12874 
12931 
12988 
13G45 



13102 
13159 
13216 
13273 
13330 



13387 
13444 
13500 
13557 
13614 



13671 
13727 
13784 
13841 
13897 



13954 
14011 
14067 
14124 
14180 



14237 
14293 
14350 
14406 
14462 



14519 
14575 
14631 
14688 
14744 



14800 
14856 
14913 
14969 
15025 



15081 
15137 
15193 
15249 
15305 



15381 
15417 
15473 
15529 
15585 



15641 



Log.Exs, 



57 
57 
57 
57 

57 
57 
57 
57 
57 

57 

5Z 
57 
57 
57 

57 
57 
57 
56 
57 

57 
57 
56 
57 
56 

57 
56 
57 
56 
56 

57 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
55 

56 



Lg, Vers. I> 



09823 
09872 
09920 
09969 
10018 

10067 
10115 
10164 
10213 
10261 



10310 
10358 
10407 
10455 
10504 



10552 
10601 
10649 
10697 

107^6 



10794 
10842 
10890 
10939 
10987 



11035 
11C83 
11131 
11179 
11227 



11275 
11323 
11371 
11419 
11467 



11515 
11562 
11610 
11658 
11706 



11754 
11801 
11849 
11897 
11944 



11992 
12039 
12087 
12134 
12182 



12229 
12277 
12324 
12371 
12419 



12466 
12513 
12560 
12608 
12655 



9-12702 
Lg. Vers, 



49 
48 
49 
48 

4? 
48 
48 
49 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
47 
48 

48 
47 
48 
48 
47 

48 

47 
47 
48 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 
47 
47 

47 



I> 



Log.Exs. I> 



15641 
15697 
15752 
15808 
15864 



15920 
15975 
16031 
16087 
16142 



16198 
16254 
16309 
16365 
16420 



16476 
16531 
16587 
16642 
16698 



16753 
16808 
16864 
16919 
16974 



17029 
17085 
17140 
17195 
17250 



17305 
17361 
17416 
17471 
17526 



17581 
17636 
17691 
17746 
17801 



17856 
17910 
17965 
18020 
18075 



18130 
18185 
18239 
18294 
18349 



18403 
18458 
18513 
18567 
18622 



18676 
18731 
18786 
18840 
18894 



18949 



Log.Exs. 



56 
55 
56 
55 

56 
55 
56 
55 
55 

56 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
54 
55 
55 
54 

55 
55 
54 
55 
54 

54 
55 
54 
54 
54 

54 
54 
55 
54 
54 

54 



10 

11 
12 
13 

li 
15 
16 
17 
18 

11 

20 
21 
22 
23 

_2i 
25 
26 
27 
28 

-29 

30 
31 
32 
33 

_34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
-49 
50 
51 
52 
53 
54 

55 
56 
57 
58 
_5i 
60 



P.P. 





57 


57 


6 


5-7 


5.7 


7 


6 


7 


6 


6 


8 


7 


6 


7 


6 


9 


8 


6 


8 


5 


10 


9 


6 


9 


5 


20 


19 


1 


19 





30 


28 


7 


28 


5 


40 


38 


3 


38 





5C 


47 


9 


47 


5 



56_ 

5-6 

6.6 

7.5 

8.5 

9-4 

18-8 

28.2 

37.6 

47.1 





56 


55 


55 


6 


5 6 


5-5 


5.5 


7 


6 


5 


6 


5 


6 


4 


8 


7 


4 


7 


4 


7 


3 


9 


8 


4 


8 


3 


8 


2 


10 


9 


3 


9 


2 


9 


1 


20 


18 


6 


18 


5 


18 


3 


30 


28 


C 


27 


7 


27 


5 


40 


37 


3 


37 





36 




50 


46 


6 

r 


46 


2 


45 


' 



6 

7 

8 

9 

10 

2t) 

30 

40 

50 



51 50 



5.4 


5. 


6 


3 


6. 


7 


2 


7. 


8 


2 


8. 


9 


1 


9. 


18 


1 


18. 


27 


2 


27- 


36 


3 


36. 


45 


4 


l45. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



51 


5-0 


5. 


5 


9 


5 


9 


5. 


6 


8 


6 


7 


6. 


7 


6 


7 


6 


7. 


8 


5 


8 


4 


8. 


17 





16 


8 


16. 


25 


5 


25 


2 


25. 


34 


C 


33 


6 


33. 


42 


5 


42 


1 


41. 



3 
2 
1 






50 


8 
6 
5 
3 
6 

3 
6 





49 


49 


48 


6 


4.9 


4 9 


4.8 


7 


5 


8 


5 


7 


5 


6 


8 


6 


6 


6 


5 


6 


4 


9 


7 


4 


7 


3 


7 


3 


10 


8 


2 


8 


1 


8 


1 


2C 


16 


5 


16 


3 


16 


1 


30 


24 


7 


24 


5 


24 


2 


40 


33 





32 




32 


3 


50 


41 


2 


40 


8 


40 


4 





48 


47 


47 


6 


4.8 


4.7 


4.7 


7 


5 


6 


5 


5 


5 


5 


8 


6 


4 


6 


3 


6 


2 


9 


7 


2 


7 


1 


7 





10 


8 





7 


9 


7 


8 


20 


16 





15 


8 


15 


6 


30 


24 





23 


7 


23 


5 


40 


32 





31 


6 


31 


3 


50 


40 





39 


6 


39 


I 



P.P. 



606 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



30' 



sr 



Lg. Vers 



9.1270_ 
12749 
12796 
12843 
12890 



12937 
12984 
13031 
13078 
13125 



13172 
13219 
13266 
13313 
13359 



13406 
13453 
13500 
13546 
13593 



13339 
13686 
13733 
13779 
13826 



13872 
13919 
13965 
14011 
14058 



14104 
14151 
14197 
14243 
14289 



14336 
14382 
14428 
14474 
145 '^i-^ 



14586 
14612 
14658 
14704 
14750 



14796 
14842 
14888 
14934 
14980 



15026 
15071 
15117 
15163 
15209 



15254 
15300 
15346 
15391 
15437 



5483 



Lg. Vers, 



Z> 



47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
46 
47 
47 
46 

47 
46 
47 
46 
46 

46 
47 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 
46 
46 
46 
46 
48 

46 
46 
46 
46 
45 

46 
45 
46 
46 
45 

45 
46 
45 
45 
45 

46 



2> 



Log.Exs. J> 



9.18949 
19003 
19058 
19112 
19167 



19221 
19275 
19329 
19384 
19438 



19492 
19546 
19601 
19655 
19709 



19763 
19817 
19871 
19925 
19979 



20033 

20087 

20141 

.20195 

.20249 



9.20303 
20357 
20411 
20465 
20518 



20572 
20628 
20680 
20733 
20787 



20841 
20894 
20948 
21002 
21055 



21109 
21162 
21216 
21289 
21323 



21376 
21430 
21483 
21537 
21590 



21643 
21697 
21750 
21803 
21857 



21910 
21963 
22016 
22070 
22123 



9-22176 



Log.Exs, 



54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

53 
54 
54 
54 
53 
54 
53 
54 
53 
54 

53 
53 
54 
53 
53 

53 
53 
53 
53 

53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 
53 



n 



Lg. Vers 



9.15483 
15528 
15574 
15619 
15665 



15710 
15755 
15801 
15846 
15891 



15937 
15982 
16027 
16073 
16118 



16163 
16208 
16253 
16298 
16343 



16388 
16434 
16479 
16523 
16568 



16613 
16658 
16703 
16748 
16793 



16838 
16882 
16997 
16972 
17017 



17061 
17106 
17151 
17195 
1724.0 



17284 
17329 
17373 
17410 
17462 



17507 
17551 
17596 
17640 
17684 



17729 
17773 
17817 
17861 
17906 



17950 
17994 
1803H 
18082 
18126 



9 •18170 
Lg. Vers. 



7> 



45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 

45 
45 
45 
45 

45 
45 
45 
44 
45 

45 
45 
45 
45 
44 

45 
44 
45 
44 
45 

44 
44 
45 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 



Log.Exs. D 



22176 
22229 
22282 
22335 
22388 

22441 
22494 
22547 
22600 
22653 



22706 
22759 
22812 
22865 
22918 



22971 
23024 
23076 
23129 
23182 



23235 
23287 
23340 
23393 
23446 



23498 
23551 
23603 
23656 
23709 



23761 
23814 
23866 
23919 
23971 



24024 
24076 
24128 
24181 
24233 



24285 
24138 
24390 
24442 
24495 



24547 
24599 
24651 
24704 
24756 



24808 
24860 
24912 
24964 
25016 



25068 
25120 
25172 
25224 
25276 



9. 25328 



Log.Exs, 



53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
52 

53 
53 
52 
53 
52 

53 
52 
53 
52 
53 

52 
52 
52 
52 
53 

52 
52 
52 
52 
52 

53 
52 
52 
52 
52 

52 
52 
52 
52 

52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 





1 
2 
3 
4 

5 
6 
7 
8 
_^ 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
_24 

25 
26 
27 
28 
29 

30 

3i 
32 
3S 

?^ 



35 
33 
37 
38 
39 

40 

41 
42 
43 

A± 

45 
46 
47 
48 
41 

50 

51 
52 
53 

54 

55 
58 
57 
58 
59 

60 



P.P. 





54 


54 


5( 


6 


5-4 


5.4 


5 


7 


6-3 


6 


3 


6 


8 


7.2 


7 


2 


7 


9 


8.2 


8 


1 


8. 


10 


9-1 


9 





8. 


20 


18.1 


18 





17. 


30 


27.2 


27 





26. 


40 


36.3 


36 





35. 


50 


45.4 


45 





44. 





53 


52 


52 


6 


5.3 


5.2 


5.2 


7 


6 


2 


6 


1 


6 





8 


7 





7 





6 


9 


9 


7 




7 


9 


7 


g 


10 


8 


8 


8 


7 


8 


6 


20 


17 


6 


17 


5 


17 


3 


30 


26 


5 


26 


2 


26 





40 


35 


3 


35 





34 


5 


50 


44 


1 


43 


7 


43 


3 



47 



6 


4 


7 


7 


5 


5 


8 


6 


3 


9 


7 


1 


10 


7 


9 


20 


15 


8 


30 


23 


7 


40 


31 


6 


50 


39 


6 



47 

7 
5 
2 




46_ 

4.6 



5 

6 

7 

7 

15 

5|23 

3 31 

1138 





46 


4 


3 


45 


6 


4-6 


4.5 


•4.5 


7 


5 


3 


5 


3 


5 


2 


8 


6 


1 


6 





6 





9 


6 


9 


6 


8 


6 


7 


10 


7 


6 


7 


6 


7 


5 


20 


15 


3 


15 


1 


15 





30 


23 


Q 


22 


7 


22 


5 


40 


30 


6 


30 


3 


30 





50 


38 


3 


37 


9 


37 


5 





44, 


6 


4.4| 


7 


5 


2 


8 


5 


9 


9 


6 


7 


10 


7 


4 


20 


14 


8 


30 


22 


2 


40 


29 


g 


50 


37 


1 



44 

4-4 

5-1 

5.8 

6.6 

7.3 

14.6 

22.0 

29.3 

36.6 



P. P. 



607 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
33° 33° 



O 

1 
2 
3 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 

ii 
20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 

21 
40 

41 
42 
43 
44 

45 
46 
47 
48 
49- 
50 
51 
52 
53 
5£ 

55 
56 
57 
58 
59 
60 



Lg. Vers. T> 



9.18170 
18214 
18258 
18302 
18346 



18390 
18434 
18478 
18522 
18566 

18610 
18654 
18697 
18741 
18785 



18829 
18872 
18916 
18959 
19003 



19047 
19090 
19134 
19177 
19221 



19264 
19308 
19351 
19395 
19438 

19481 
19525 
19568 
19611 
19654 



19698 
19741 
19784 
19827 
19870 



19914 
19957 
20000 
20043 
20086 



20129 
20172 
20215 
20258 
20301 



20343 
20386 
20429 
20472 
20515 



20558 
20600 
20643 
20686 
20728 



9.;:>n771 



Lg.Vers. T> 



44 
44 
44 
44 

44 
44 
44 
44 
43 

44 
44 
43 
44 
43 

44 
43 
43 
43 
44 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

42 
43 
43 
43 
42 

43 
42 
43 
43 
42 

43 



Log.Exs. -Z> Lg. Vers. -D Log.Exs. 



25328 
25380 
25432 
25484 
25536 

25588 
25640 
25692 
25743 
25795 



25847 
25899 
25950 
26002 
26054 



26105 
26157 
26209 
26260 
26312 



26364 
26415 
26467 
26518 
26570 



26621 
26673 
26724 
26776 
26827 



26878 
26930 
26981 
27032 
27084 



27135 
27186 
27238 
27289 
27340 



27391 
27443 
27494 
27545 
27596 



27647 
27698 
27749 
27800 
27852 



27903 
27954 
28005 
28056 
28107 



28157 
28208 
28259 
28310 
28361 



9. 28412 



Log, Exs< 



2> 



20771 
20814 
20856 
20899 
20942 

20984 
21027 
21069 
21112 
21154 



21196 
21239 
21281 
21324 
21366 



21408 
21451 
21493 
21535 
21577 



21620 
21662 
21704 
21746 
21788 



21830 
21872 
21914 
21956 
21998 



22040 
22082 
22124 
22166 
22208 



22250 
22292 
22334 
22376 
22417 



22459 
22501 
22543 
22584 
22626 



22668 
22709 
22751 
22792 
22834 



22876 
22917 
22959 
23000 
23042 



23083 
23124 
23166 
23207 
232 48 
9.23290 



Lg. Vers 



42 
42 
42 
43 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
4l 
42 
42 
4l 

42 
4l 
42 
4l 
4l 

42 
4l 
4T 
4l 
41 

4l 
41 
4l 
4l 
4l 

41 
41 
4l 
41 
41 

4l 



9.28412 
28463 
28514 
28564 
28615 



28666 
28717 
28768 
28818 
28869 



28920 
28970 
29021 
29072 
29122 



29173 
29223 
29274 
29324 
29375 



29426 
29476 
29527 
29577 
29627 



29678 
29728 
29779 
29829 
29879 



29930 
29980 
30030 
30081 
30131 



30181 
30231 
30282 
30332 
30382 



30432 
30482 
30533 
30583 
30633 



30683 
30733 
30783 
30833 
30883 



30933 
30983 
31033 
31083 
31133 



31183 
31233 
31283 
31333 
31383 



31432 



Log.Exs. 



2> 

51 
51 
50 
51 

51 
50 
51 
50 
50 

51 
50 
50 
51 
50 

51 
50 
50 
50 
51 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

49 
50 
50 
50 
50 

49 

17 





1 

2 
3 
4 

5 
6 
7 
8 
9 

10 

II 
12 
13 
14 

15 
16 
17 
18 
Jl 
20 
21 
22 
23 
24 

25 
26 
27 
28 
-29 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
38 
59 

60 



P.P. 





52 


^i, 


6 


5-2 


5.11 


7 


6 





6 





8 


6 


9 


6 


8 


9 


7 


8 


7 


7 


10 


8 


6 


8 


6 


20 


17 


3 


17 


1 


30 


26 





25 


7 


4C 


34 


6 


34 


3 


50 


43 


3 


42 


9 



50_ 

50' 



50 

50 



816 
225 
633 
ll41 



51 

51 

5.9 

6.8 

7-6 

8.5 

17.0 

25.5 

34.0 

42.5 



49 

4.9 





44 


43 


4J 


6 


4.4 


4.3 


4. 


7 


5 


1 


5 


1 


5. 


8 


5 


g 


5 


8 


5. 


9 


6 


6 


6 


5 


6. 


10 


7 


3 


7 


2 


7- 


20 


14 


6 


14 


5 


14. 


30 


22 





21 


7 


21. 


40 


29 


3 


29 





28. 


50 


36 


6 


36 


2 


35. 





42 


42 


6 


4.2 


4.2! 


7 


4.9 


4 


9 


8 


5-6 


5 


6 


9 


6.4 


6 


3 


10 


7.1 


7 





20 


14.1 


14 





30 


21.2 


21 





40 


28.3 


28 





50 


35.4 


35 






41_ 

41 

4.8 

5.5 

6.2 

6.9 

13. 8 

20.7 

27-6 

34.6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



41 

4.1 
48 

5.4 
6.1 
6.8 
13.6 
20.5 
27.3 
34.1 



P. P. 



608 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
34'' 35° 



Lg.Vers, 



23290 
23331 
23372 
23414 
23455 



23496 
23537 
23579 
23620 
23661 



23702 
23743 
23784 
23825 
23866 



23907 
23948 
23989 
24030 
24071 



24112 
24153 
24194 
24235 
24275 



24316 
24357 
24398 
24438 
24479 



24520 
24561 
24601 
24642 
24682 



24723 
24764 
24804 
24845 
24885 



24926 
24966 
25007 
25047 
25087 



25128 
25168 
25209 
25249 
25289 



25329 
25370 
25410 
25450 
25496 



25531 
25571 
25611 
25651 
25691 



9-25731 



Lg. Vers. 



D 



n 



Log.Exs 



9.31432 
31482 
31532 
31582 
31632 



31681 
31731 
31781 
31831 
31880 



31930 
31980 
32029 
32079 
32129 



32178 
32228 
32277 
32327 
32377 



32426 
32476 
32525 
32575 
32624 



32673 
32723 
32772 
32822 
32871 



32920 
32970 
33019 
33069 
33118 



33167 
33216 
33266 
33315 
33364 



33413 
33463 
33512 
33561 
33610 



33659 
33708 
33758 
33807 
33856 



33905 
33954 
34003 
34052 
34101 



34150 
34199 
34248 
34297 
34346 



34395 



Log.Exs. 



J> 



50 
50 
49 
50 

49 
50 
49 
50 
49 
50 
49 
49 
49 
50 

49 
49 
49 
49 
50 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 

48 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
.49 

49 



n 



Lg. Vers. 



25731 
25771 
25811 
25851 
25891 



25931 
25971 
26011 
26051 
26091 



26131 
26171 
26210 
26250 
26290 



26330 
26370 
26409 
26449 
26489 



26528 
26568 
26608 
26647 
26687 



26726 
26766 
26806 
26845 
26885 



26924 
26964 
27003 
27042 
27082 



27121 
27161 
27200 
27239 
27278 



27318 
27357 
27396 
27435 
27475 



27514 
27553 
27592 
27631 
27670 



27709 
27749 
27788 
27827 
27866 

27905 
27944 
27982 
28021 
28060 



9-28099 
Lg. Vers, 



I> 



40 
40 
40 
40 

40 
40 
40 
39 
40 

40 
40 
39 
40 
40 

39 
40 
39 
40 
39 

39 
40 
39 
39 
39 

39 
40 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
38 
39 
39 

39 



Log.Exs. 



9-34395 
34444 
34492 
34541 
34590 



34639 
34688 
34737 
34785 
34834 



34883 
34932 
34980 
34029 
35078 



35127 
35175 
35224 
35273 
35321 



35370 
35419 
35467 
35516 
35564 



35613 
35661 
35710 
35758 
35807 



35855 
35904 
35952 
36001 
36049 



36098 
36146 
36194 
36243 
3629T 



36340 
36388 
36436 
36484 
36533 



36581 
36629 
36678 
36726 
36774 



36822 
36870 
36919 
36967 
37015 



37063 
37111 
37159 
37207 
37255 



9- 37303 



Log.Exs. 





1 
2 
3 
4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
-29 
30 
31 
32 
33 

35' 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
_49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



P.P. 





50 


49 


49 


6 


5.0 


4.9 


4.9 


7 


5 


3 


5 


8 


5 


7 


8 


6 


6 


6 


6 


6 


5 


9 


7 


5 


7 


4 


7 


3 


10 


8 


3 


8 


2 


8 


1 


20 


16 


6 


16 


5 


16 


3 


30 


25 





24 


7 


24 


5 


40 


33 


3 


33 





32 


6 


50 


41 


6 


41 


2 


40 


8 



48 

4.8 
6 
4 
3 
1 



30 24 
40i32 
50140 



48 
4-8 



5 
6 
7 
8 

1116 
2:24 
3132 
4] 40 



41_ 

4-1 



41 

4.1 



6;27 
6134 





40 


40 


6 


4.0 


4.0 


7 


4 


7 


4 


6 


8 


5 


4 


5 


3 


9 


6 


1 


6 





10 


6 


7 


6 


6 


20 


13 


5 


13 


3 


30 


20 


2 


20 





40 


27 





26 


6 


50 


33 


7 


33 


3 



39 


31 


3.9 


3. 


4.6 


A. 


5.2 


5- 


5.9 


5. 


6.6 


6- 


13.1 


13. 


19-7 


19. 


26.3 


26. 


32. 9 


32. 



38 

3.3 

4.5 

5.1 

5.8 

6.4 

12-8 

19.2 

25-6 

?2.1 



P. P. 



009 



TABLE VIIL—LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
36° 37° 



Lg. Vers. 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 

21 
40 

41 
42 
43 

/'A. 



'1 

ro 

Gl 

I o 
r 3 
54 

55 
56 
57 
58 
59 

60 



9.28099 
.28138 
.28177 
.28816 
•28255 



9-28293 
•28332 
.28371 
.28410 
•28448 



9-28487 
28526 
28564 
28603 
28642 



9^28680 
.28719 
.28757 
.28796 
•28835 



9 28873 
.28912 
.28950 
.28988 
•29027 



9 29065 
•29104 
.29142 
.29180 
29219 



9.29257 
.29295 
.29334 
.29372 
•2941C 



9 • 29448 
.29487 
.29525 
.29563 
•29601 



9 •2963? 
.29677 
•29715 
•29754 
.29792 



9.29830 
29868 
29906 
29944 
29982 



9.30020 
.30057 
.30095 
.30133 
•30171 



9.30209 
.30247 
.30285 
.30322 
.30360 



9-30398 
Lg. Vers. 



2> 

3? 
38 
39 
39 

38 
39 
38 
3? 
38 
3? 
38 
38 
39 
38 

38 
38 
38 
38 
39 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
37 
38 
38 
38 

38 
37 
38 
37 
38 

38 



Log.Exs, 



37303 
37352 
37400 
37448 
37496 



37544 
37592 
37640 
37687 
37735 

37783 
37831 
3787S 
37927 
37975 



38023 
38071 
38119 
38166 
38214 



38262 
38310 
38357 
38405 
38453 



38501 
38548 
38596 
38644 
38692 



38739 
38787 
38834 
38882 
38930 

38977 
39025 
39072 
39120 
39168 



39215 
39263 
39310 
39358 
39405 



39453 
39500 
39548 
39595 
39642 



39890 
39737 
39785 
39832 
39879 



39927 
39974 
40021 
40069 
40116 



40163 



Log.Exs, 



2> 

48 
48 
48 
48 

48 
48 
48 
47 
48 

48 
48 
48 
48 

47 

48 
48 
48 
47 
48 

47 
48 
47 
48 
47 
48 
47 
48 
47 
48 

47 
47 
47 
48 
47 

47 
47 
47 
48 
47 

47 
48 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 

4Z 
47 

47 
47 
47 
47 
47 

47 



Lg. Vers. 



9-30398 
.30436 
.30474 
.30511 
-30549 



-30587 
.30624 
.30662 
.30700 
-30737 



9-30775 
•30812 
.30850 
.30887 
•309 25 

9 • 30962 
.31000 
.31037 
.31075 
•31112 



9.31150 

.31187 

.31224 

•31262 

31299 



9-31336 
.31374 
.31411 
.31448 
•31485 



9^31523 
.31560 
.31597 
.31634 
.31671 



9.31708 
.31746 
.31783 
.31820 
•31857 



9^31894 
.31931 
.31968 
.32005 
- P2042 



9.32079 
.32116 
.32153 
.32190 

. .32227 

9.32263 
.32300 
.32337 
.32374 
.32411 



9 . 32447 
. 32484 
.32521 
.32558 
.32594 



9-32631 



Lg. Vers. 



1> 

37 
38 
37 
37 

38 

37 
37 
38 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

36 
37 
37 
36 
37 

36 
37 
36 
37 
36 
37 



Log.Exs. 



9.40163 
.40210 
.40258 
.40305 
-40352 



9.40399 
. 40447 
• 40494 
. 4C541 
-40588 

9-40635 
•40682 
•40730 
•40777 
40824 



9-40871 
40918 
40965 
41012 
4IC59 



9-41106 
41153 
412C0 
41247 
41294 



9-41341 

•41383 

•41435 

•41482 

41529 



41576 
41623 
41670 
41717 
41763 



•41810 
•41857 
•41904 
•41951 
41998 



9 . 42044 
•42091 
.42138 
.42185 
.42231 



9.42278 
.42325 
.42372 
.42418 
•42465 



9.42512 
.42558 
.42605 
.42652 
•42698 



9.42745 
.42792 
.42838 
.42885 
.42931 



9-42978 



|Log. 



Exs, 



I> 

47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

46 
47 
47 
47 
46 

47 
47 
46 
47 
47 
46 
47 
46 
47 
46 

47 
46 
47 
46 
47 

46 
46 
47 
46 
46 

46 
47 
46 

46 
46 
46 



5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 

ii 
20 

21 
22 
23 
24 

25 
26 
27 
28 
_29 

30 

31 
32 
33 
-34 

35 
36 
37 
38 
-39 
40 
41 
42 
43 
44 

45 

46 

47 

48 

_49 

50 

51 

52 

53 

_54 

55 
56 
57 
58 
_59 

60 



P. P. 



48 



6 
7 
8 

9 
10 
20 
30 
40 
50 



4 8f 


5 


6 


6 


4 


7 


3 


8 


1 


16 


1 


24 


2 


32 


3 


40 


4 



47 



4 


7 


4. 


5 




5^ 


6 


3 


6. 


7 


1 


7. 


7 


9 


7. 


15 


8 


15. 


23 


7 


23. 


31 


6 


31. 


39 


6 


39. 



48 

48 

56 

6.4 

7.2 

80 

160 

24.0 

32. 

40.0 

47 
7 
5 
2 
Q 
8 
6 
5 
3 

i 



6 

7 
8 
9 
10 
20 
30 
40 
50 



46_ 

4.6 

5.4 

6.2 

7.0 

7.7 

15.5 

23.2 

31.0 

38.7 





39 


6 


3-9 


7 


4-5 


8 


5.2 


9 


5.8 


10 


6.5 


20 


13.0 


30 


10.5 


40 


26.0 


50 


32.5 



6 

7 
8 
9 
10 
20 
30 
40 
50 



38 

38 

4.4 

5.0 

5.7 

6.3 

12.6 

19.0 

25.3 

31.6 





37 


6 


3.7 


7 


4.3 


8 


4.9 


9 


5.5 


10 


6.1 


20 


12.3 


30 


18.5 


40 


24.6 


PP 


30.8 



38 

3.8 

4.5 

5.1 

5.8 

6.4 

12.8 

19.2 

25.6 

32.1 

^^;, 

3.7 

4.4 

5.0 

5.6 

6.2 

12.5 

18.7 

25.0 

31.2 

36 

3.6 

4-2 

4.8 

5.5 

6.1 

12.1 

18-2 

24.3 

30.4 



P.P. 



610 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
38*" 39° 



Lg. Vers. 



32631 
32668 
32704 
32741 
32778 



32814 
32851 
32888 
32924 
32961 



32997 
33034 
33070 
33107 
33143 



33180 
33216 
33252 
33289 
33325 



33361 
33398 
33434 
33470 
33507 



33543 
33579 
33615 
33652 
33688 



33724 
33760 
33796 
33833 
33869 



33905 
33941 
33977 
34013 
34049 



34085 
34121 
34157 
34193 
34229 



34265 
34301 
34337 
34373 
34408 
34444 
34480 
34516 
v34552 
34587 

34623 
34659 
34695 
3473Q 
347Rfi 



9. 34802 



Lg.Vers. 



Log.Exs, 



42978 
43024 
43071 
43118 
43164 



43211 
43257 
43304 
43350 
43396 



43443 
43489 
43536 
43582 
43629 



43675 
43721 
43768 
43814 
43861 



43907 
43953 
43999 
44046 
44092 



44138 
44185 
44231 
44277 
44323 



44370 
44416 
44462 
44508 
44554 



44601 
44647 
44693 
44739 
44785 



44831 
44877 
44924 
44970 
45016 



45062 
45108 
45154 
45200 
45246 



45292 
45338 
45384 
45430 
45476 



45522 
45568 
45614 
45660 
45706 



^5752 



X> Log.Exs. 



46 
47 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 



Lg. Vers, 



34802 
34837 
34873 
34909 
34944 



34980 
35016 
35051 
35087 
35122 



35158 
35193 
35229 
35264 
35300 

35335 
35370 
35406 
35441 
35477 



35512 
35547 
35583 
35618 
35653 



35689 
35724 
35759 
35794 
35829 



35865 
35900 
35935 
35970 
36005 



36040 
36076 
36111 
36146 
36181 



36216 
36251 
36286 
36321 
36356 



36391 
36426 
36461 
36495 
33530 



36565 
36600 
36635 
36670 
36705 



3673P 
36774 
36809 
3684fi 
3fi87P 



O.^'^'.Pl.'^ 



Lg. Vers. 



35 
36 
35 
35 

35 
36 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 
35 
35 
35 
35 
35 

35 
35 
35 
35 
35 
35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
34 
35 

35 
35 
34 
35 
35 

34 
35 
34 
35 
34 
35 



Loof.Exs 



45752 
45797 
45843 
45889 
45935 



4598 

46027 

46073 

46118 

46164 



46210 
46256 
46302 
46347 
46393 



46439 
46485 
46530 
46576 
46622 



46668 

46713 

46759 

4680 

46850 



46896 
46942 
46987 
47033 
47078 



47124 
47170 
47215 
47261 
47306 



47352 
47398 
47443 
47489 
47534 



47580 
47625 
47671 
47716 
47762 



47807 
47852 
47898 
47943 
47989 



48034 
48080 
48125 
48170 
4R21R 



48^61 
48306 
4835? 
483P7 



/■RA<?R 



J) Log.Exs. n 



2> 

45 
46 
4S 
46 

45 
46 
46 
45 
46 

46 
45 
46 
45 
46 

45 
46 
45 
46 
45 

46 
45 
45 
46 
45 
45 
46 
45 
45 
45 

46 
45 
45 
45 
45 

46 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
_24 

25" 
26 
27 
28 
29 

30 

31 
32 
33 
1± 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
^ 
GO 



P.P. 



47 



4 


7 


4. 


5 


5 


5. 


6 


2 


6. 


7 





7. 


7 


3 


7. 


15 


6 


15. 


23 


5 


23- 


31 


3 


31- 


39 


1 


38. 



46 

4. 61 



7j 5 
8 6 
9, 6 
10 7 
20 15 
30,23 
4030 
50138 



46_ 

6 



45 

4.5 



6 


4. 


7 


5. 


8 


6. 


9 


6. 


10 


7. 


20 


15- 


30 


22. 


40 


30. 


50 


37. 



3| 5 
1 6 
9 6 
6 7 
3'l5 
0'22 
6;30 
3137 

45 

5 
2 

7 
5 

5 

5 



e 

7 
P 

e 

10 
20 
30 
40 



37 



3 


7 


3. 


4 


3 


4. 


4 


9 


4. 


5 


5 


5. 


6 


1 


6. 


12 


3 


12. 


18 


5 


18. 


24 


6 


24. 


30 


8 


30. 



36 

3.6 



4.2 
4.8 
5.4 
60 



12.0 
18.0 
24.0 
30.0 



36_ 

6 
2 
8 
5 
1 
1 
2 
3 
4 

35 

3-5 



35 


3? 


3.5 


3.4 


4.1 


4 





4.6 


4 


6 


5.2 


5 


2 


5.8 


5 


7 


11-6 


11 


5 


17.^ 


17 


2 


23-.^ 


23 





l?Q.l 


28 


7 



P.P. 



611 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
40° 41° 





1 
2 
3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

20 

21 
22 
23 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
•47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Lg. Vers. 



36913 
36948 
36982 
37017 
37052 



37086 
37121 
37156 
37190 
37225 



37259 
37294 
37328 
37383 
37397 



37432 
37466 
37501 
37535 
37570 



37604 
37639 
37673 
37707 
37742 

37776 
37810 
37845 
37879 
37913 



37947 
37982 
38016 
38050 
38084 



38118 
38153 
38187 
38221 
38255 



38289 
38323 
38357 
38391 
38425 



38459 
38493 
38527 
38561 
38595 



38629 
38663 
38697 
38731 
38765 



38799 
38833 
38866 
38900 
38934 



9.3«Pfift 



Lg. Vers. 



2> 



34 
34 
35 
34 
34 
35 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
33 
34 

34 
34 
33 
34 
33 
34 



Log.Exs, 



9.48488 
.48533 
•48578 
.48624 
•48669 



9-48714 
•48759 
•48805 
•48850 
-48895 



9^48940 
.48986 
.49031 
.49076 
•49121 



9.49166 
•49211 
•49257 
•49302 
-49347 



9-49392 
.49437 
.49482 
.49527 
-49572 



9.49618 
•49663 
•49708 
•49753 
•49798 



9.49843 
•49888 
•49933 
.49978 
-50023 



•50068 
•50113 
•50158 
•50203 
•50248 



9-50293 
•50338 
•50383 
•50427 
-50472 



9-50517 
•50562 
•50607 
•50652 
-50697 



9-50742 
50787 
50831 
50876 
50921 



9-50966 
•5101T 
•51055 
-51100 
-51145 



9-51190 
Log.Exs 



Lg.Vers. 



45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
44 
45 

45 
45 
45 
44 
45 

45 
45 
44 
45 
45 

44 
45 
44 
45 
45 

44 



38968 
39002 
39035 
39069 
39103 



39137 
39170 
39204 
39238 
39271 



39305 
39339 
39372 
39406 
39439 



39473 
39507 
39540 
39574 
39607 



39641 
39674 
39708 
39741 
39774 



39808 
39841 
39875 
39908 
39941 



39975 
40C08 
40041 
40075 
40108 



40141 
40175 
40208 
40241 
40274 



40307 
40341 
40374 
40407 
40440 



40473 
405C6 
40540 
40573 
40606 



40639 
4G672 
40705 
40738 
40771 



40804 
40837 
40870 
40903 
40936 



409B9 



JD Lg.Vers, 



2> 

34 
33 
34 
33 

34 
33 
33 
34 
33 
33 
34 
33 
33 
33 

33 
34 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 
33 
33 
33 
33 
33 

33 
23 
33 
33 
33 

33 
33 

3^ 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 
33 



.og.Exs, 



51190 
51235 
51279 
51324 
51369 



51414 
51458 
51503 
51548 
51592 



51637 
51682 
51726 
51771 
51816 



51860 
51905 
51950 
51994 
52039 



52084 
52128 
52173 
52217 
52262 



2> 



52306 
52351 
52396 
52440 
52485 



52529 
52574 
52618 
52663 
52707 



52752 
52796 
52841 
52885 
52930 



52974 
53018 
53063 
53107 
53152 



53196 
53240 
53285 
53329 
53374 



53418 
53462 
53507 
53551 
53595 



53640 
53684 
53728 
53773 
53817 



9.53RP1 



Log.Exs 



45 
44 
45 
44 

45 
44 
45 
44 
44 

45 
44 
44 
45 
44 

44 
45 
44 
44 
44 

45 
44 
44 
44 
44 
44 
45 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 

u 





1 

2 
3 
j4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 



20 

21 
22 
23 
_24 

25 
26 
27 
28 
-29 

30 

31 

32 

33 

_34 

35 
36 
37 
38 
39 



40 

41 
42 
43 

A^ 
45 
46 
47 
48 

_49 

50 

51 

52 

53 

_54 

55 
56 
57 
58 
59 

60 



P.P. 



45, 


4 51 


5 


3 


6 





6 


8 


7 


6 


15 


1 


22 


7 


30 


3 


37 


9 





43 


6 


4-41 


7 


5 


2 


8 


5 


9 


9 


6 


7 


10 


7 


4 


20 


14 


8 


30 


22 


2 


40 


29 


6 


50 


37 


1 



35 


8 


5 


4 


1 


4 


6 


5 


2 


5 


8 


11 


6 


17 


5 


23 


3 


29 


1 





34 


6 


3^4| 


7 


3 


9 


8 


4 


5 


9 


5 


1 


10 


5 


6 


20 


11 


3 


30 


17 





40 


22 


6 


50 


28 


3 



P. p. 



612 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



4.2' 



43' 



O 

1 

2 

3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 



20 

21 

22 

23 

21 

25 

26 

27 

28 

21 

30 

31 

32 

33 

31 

35 

36 

37 

38 

39 



40 

41 

42 

43 

^1 

45 

46 

47 

48 

49^ 

50 

51 
52 
53 
51 
55 
56 
57 
58 
51 
60 



Lg. Vers 



40969 
41001 
41034 
41067 
41100 



41133 
41166 
41199 
41231 
41284 



41297 
41330 
41362 
41395 
41428 



41481 
41493 
41526 
41559 
41591 



41624 
41657 
41689 
41722 
41754 



41787 
41819 
41852 
41885 
41917 



41950 
41982 
42014 
42047 
42079 



42112 
42144 
42177 
42209 
42'34T 



42274 
42306 
42338 
42371 
42403 



42435 
42467 
42500 
42532 
42564 



42596 
42629 
42661 
42693 
42725 



42757 
42789 
42822 
42854 
42886 



9-42918 



Lg. Vers. 



7> 



Log.Exs 



9-53861 
53906 
53950 
53994 
54038 



54083 
54127 
54171 
54215 
54259 



54304 
54348 
54392 
54436 
54480 



54525 
54569 
54813 
54657 
54701 



54745 
54790 
54834 
54878 
54922 



54966 
55010 
55054 
55098 
55142 



55186 
55230 
55275 
55319 
55363 



55407 
55451 
55495 
55539 
55583 



55627 
55671 
55715 
55759 
55803 



55847 
55890 
55934 
55978 
56022 



56066 
56110 
56154 
56193 
56242 



56286 
56330 
56374 
56417 
56461 



9-56505 
Log.Exs. 



44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 
43 
44 
44 
44 
44 
44 
44 
43 
44 

44 
44 
44 
43 
44 

43 



Lg. Vers, 



9-42918 
42950 
42982 
43014 
43046 



43078 
43110 
43142 
43174 
432C6 



43238 
43270 
43302 
43334 
43355 



43397 
43429 
43461 
43493 
43525 



43557 
43588 
43620 
43652 
43684 



43715 
43747 
43779 
43810 
43842 



43874 
43906 
43937 
43969 
44000 



44032 
44064 
44095 
44127 
44158 



44190 
44221 
44253 
44284 
44316 



44347 
44379 
44410 
44442 
44473 



44504 
44536 
44567 
44599 
44630 



44661 
44693 
44724 
44755 
44787 



9-44818 
Lg. Vers. 



I) 



32 
32 
32 
32 

32 
32 
31 
32 
32 

32 
32 
32 
32 
31 

32 
32 
32 
31 
32 

32 
31 
32 
31 
32 

31 
32 
31 
31 
32 

31 
32 
31 
31 
31 

32 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 



Log.Exs. 



56505 
56549 
56593 
56637 
56680 



58724 
56768 
56812 
56858 
56899 



56943 
58987 
57031 
57075 
57118 



57162 
57206 
57250 
57293 
57337 



57381 
57424 
57468 
57512 
57556 



57599 
57643 
57687 
57730 
57774 



57818 
57881 
57905 
57949 
57992 



58036 
58079 
58123 
58167 
58210 



58254 
58297 
58341 
58385 
58428 



58472 
58515 
58559 
58602 
58646 



58689 
58733 
58776 
58820 
58864 



58907 
58951 
58994 
59037 
59081 



59124 



» 



43 
44 
44 
43 

44 
44 
43 
44 
43 

44 
44 
43 
44 
43 

44 
43 
44 
43 
44 

43 

43 
44 
43 
44 

43 
43 
44 
43 
43 

44 
43 
43 
44 
43 

43 
43 
44 
43 
43 

43 
43 
44 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
44 
43 

43 
43 
43 
43 
43 

43 



J> Log.Exs. 2> 
~613 



io 

11 

12 
13 
JA 
15 
18 
17 
18 
_19 

20 

21 
22 
23 
24 
25 
26 
27 
28 
21 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
41 
50 
51 
52 
53 
54 

55 
56 
57 
58 
51 
60 



P.P. 



44_ 

4.4: 

5.2! 

5.9; 

6.7 

7.4' 
14.8|14 
22.2:22 
29.6i29 
37.l'36 



44 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 
7 
8 
9 

10 
20 
30 
40 
50 



32 

3.2 
3.8 

4.3 
4.9 
5.4 
10-8 
16.5:16.2 
22.021-6 
27.5127.1 





43 


4^ 


6 


4.3 


4- 


7 


5.1 


5. 


8 


58 


5. 


9 


6-5 


6. 


10 


7.2 


7. 


20 


14-5 


14. 


30 


21-7 


21. 


40 


29.0 


28. 


50 


36.2 


35. 



33 

3.3 
3-8 
4.4 
4.9 
5.5 
11.0 



32 



3 


2 


3. 


3 


7 


3. 


4 


2 


4. 


4 


8 


4. 


5 


3 


5. 


10 


6 


10. 


16 





15. 


21 


3 


21. 


26 


6 


26. 



31_ 

1 
7 
2 
7 
2 
5 
7 

2 



31 

3.1 



20 10 
30 15 
40 20 
50I25 



P. P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



44^ 



45^ 



Lg.Vers. n 



9.44818 
•44849 
•44880 
•44912 
•44943 



5 
6 
7 
8 
_9^ 

10 

11 

12 

13 

Ik 

15 

16 

17 

18 

il 

20 

21 

22 

23 

21 

25 

26 

27 

28 

29_ 

30 

31 

32 

33 

34 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49^ 

50 

51 

52 

53 

51 

55 

56 

57 

58 

59. 

60 



9-44974 
•45005 
•45036 
•45068 
•45099 



45130 
45161 
45192 
45223 
45254 



45285 
45316 
45348 
45379 
45410 



45441 
45472 
45503 
45534 
45585 



45595 
45626 
45657 
45688 
45719 



45750 
45781 
45812 
45843 
45873 



45904 
45935 
45966 
45997 
46027 



46058 
46089 
48120 
46150 
46181 



46212 
46242 
46273 
46304 
46334 



9-46365 
46396 
46426 
40457 
46487 



9-46518 

-46549 

-46579 

43610 

46640 



9. 46671 



' Lg. Vers, 



31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

30 
31 
31 
31 
31 

31 
30 
31 
31 
30 

31 
31 
30 
31 
30 

31 
30 
31 
30 
31 

30 
30 
31 
30 
30 

31 
30 
30 
30 
30 

31 
30 
30 
30 
30 
30 

2> 



Log.Exs. 



59124 
59168 
59211 
59255 
59298 



59342 
59385 
59429 
59472 
59515 



59559 
59602 
59646 
59689 
59732 



59776 
59819 
59863 
59906 
59949 



59993 
60036 
60079 
60123 
60166 



60209 
60253 
60296 
60339 
60383 



60426 
60469 
60512 
60556 
60599 



60642 
60685 
60729 
60772 
60815 

60858 
80902 
60945 
60988 
61031 



61075 
61118 
61161 
61204 
61247 



61291 
61334 
61377 
61420 
61463 



61506 
61550 
61593 
61636 
61679 



61722 



|Log.Exs 



43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 

43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 



Lg. Vers. 



9-46671 
46701 
46732 
46762 
46793 



9-46823 
48853 
46884 
46914 
46945 



9-46975 
47005 
47036 
47066 
47096 



9-47127 
-47157 
-47187 
-47218 
-47248 



9-47278 
-47308 
-47339 
•47369 
•47399 



9-47429 
-4745S 
-4749C 
-47520 
-47550 



JO 



9-47580 
47610 
47640 
47670 
47700 



9-47731 
•47761 
•47791 
-47821 
•^7851 

9-47881 
-47911 
-47941 
-47971 
-48001 



9-48031 
•48061 
•48090 
•48120 
-48150 



IJ 



9-48180 
-48210 
-48240 
•48270 
-48300 

9-48320 
•48359 
•48389 
48419 
48449 



9 -48478 
Lg. Vers 



30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 

30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
29 
30 
30 

30 
30 
29 
30 
30 

29 
30 
30 
29 
30 
29 

n5 



Log.Exs. 



9-61722 
•61765 
•61808 
•61852 
•61895 



9-61938 
•61981 
•62024 
•62067 
-62110 



9-62153 
■62196 
-62239 
•62282 
-62326 



9-62369 
-62412 
•62455 
•62498 
-62541 



9^62584 
-62627 
-62670 
•62713 
-62756 



62799 
62842 
62885 
62928 
62971 



9-63014 
•63057 
•63100 
•63143 
-63186 



63229 
63272 
63315 
63358 
63401 



9 - 63443 

• 63486 

•63529 

-63572 

63615 



9-63658 
63701 
63744 
63787 
63830 



9 •63873 
•63915 
•63958 
• 64001 
- 64044 



9-64087 
.64130 
•64173 
.64216 
•64258 



9-64301 



Log.ExSi 



D 

43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43 

43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

42 
43 
43 
43 
43 

43 
42 
43 
43 
43 

43 
42 
43 
43 
43 

42 
43 
43 
43 
42 

43 



10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

30 

21 
22 
23 
24 



30 

31 
32 
33 
34 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 



50 

51 
52 
53 
54 



55 
56 
57 
58 
_59 
60 



P. P. 



43_ 

4-3 



43 

4-3 



6 


4. 


7 


4- 


8 


5^ 


9 


6^ 


10 


1- 


20 


14 • 


30 


21 • 


40 


28. 


50 


35. 



43 

2 
9 
6 
4 
1 
I 
2 
3 
4 



31 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



31 

3^1 



30, 


3-01 


3 


5 


4 





4 


6 


5 


1 


10 


1 


15 


2 


20 


3 


25 


4 



30 

30 

3.5 

4.0 

4.5 

5.0 

10^0 

15. 

20^0 

25.0 



29 

2.9 

3.4 

3-9 

4.4 

4.9 

9.8 

14.7 

19.6 

24.6 



P. P. 



614 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



46^ 



47* 



Lg. Vers. 



9.48478 
48508 
48538 
48568 
48597 



48627 
48657 
48686 
48716 
48746 



48775 
48805 
48835 
48864 
48894 



48923 
48953 
48983 
49012 
4904? 



49071 
49101 
49130 
49160 
49189 



49219 
49248 
49278 
49307 
49336 



49300 
49395 
49425 
49454 
49483 



49513 
49542 
49571 
49601 
49630 



49059 
49689 
49718 
49747 
49776 



49806 
49835 
49864 
49893 
49922 

49952 
49981 
50010 
50039 
50068 



50097 
50126 
50155 
50185 
50214 



50243 



' Lg. Vers 



1} 



30 
29 
30 
29 

30 
29 
29 
30 
29 

29 
30 
29 
29 
29 

29 
30 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
2C 
29 

29 



Log.Exs, 



9.64301 
64344 
64387 
64430 
64473 



64515 
64558 
64601 
64644 
64687 



64729 
64772 
64815 
64858 
64901 



64943 
64986 
65029 
65072 
65114 



65157 
65200 
65243 
65285 
65328 



65371 
65414 
65456 
65499 
65542 



65585 
65627 
65670 
65713 
65755 



65798 
65841 
65884 
65926 
65969 



66012 
66054 
66097 
66140 
66182 



66225 
66268 
66310 
66353 
66396 



66438 
66481 
66523 
66566 
66609 



66651 
66694 
66737 
66779 
66822 
66864 
Log.Exs. 



43 
42 
43 
43 

42 
43 
43 
42 
43 
42 
43 
43 
42 
43 
42 
43 
42 
43 
42 

43 

42 
43 
42 
43 

42 
43 
42 
43 
42 

43 
42 
43 
42 
42 

43 
42 
43 
42 
42 

43 
42 
42 
43 
42 

42 
43 
42 
42 
43 

42 
42 
42 
43 
42 

42 
42 
43 
42 
42 

42 



La:. Vers. 



9.50243 
50272 
50301 
50330 
50359 



50388 
50417 
50446 
50475 
50504 



50533 
50562 
50591 
50619 
50648 



50677 
50706 
50735 
50764 
50793 



50821 
50850 
50879 
50908 
50937 



50965 
50994 
51023 
51052 
51080 



51109 
51138 
51167 
51195 
51224 



51253 
51281 
51310 
51338 
51367 



51398 
51424 
51453 
51481 
51510 



51539 
51567 
51596 
51624 
51653 



51681 
51710 
51738 
51767 
51795 



51823 
51852 
51880 
51909 
51937 
51965 
Lg. Vers. 



2> 



29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
28 
29 

29 
29 
28 
29 
29 

28 
29 
29 
28 
29 

28 
29 
28 
29 
28 
29 
28 
29 
28 
28 

29 
28 
28 
28 
29 

28 
28 
28 
28 
28 

29 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 



Log.Exs. 



66864 
66907 
66950 
66992 
67035 



67077 
67120 
67162 
67205 
67248 



67290 
67333 
67375 
67418 
67460 



67503 
67546 
67588 
67631 
67673 



1} 



67716 
67758 
67801 
67843 
67886 



67928 
67971 
68013 
68056 
68098 



68141 
68183 
68226 
68268 
68311 



68353 
68396 
68438 
68481 

68523 



68566 
68608 
68651 
68693 
68735 



68778 
68820 
68863 
68905 
68948 



68990 
69033 
69075 
69117 
69160 



69202 
69245 
69287 
69330 
69372 



69414 
Log.Exs. 



42 
43 
42 
42 

42 
42 
42 
43 
42 

42 
42 
42 
42 
42 
42 
43 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 



30 

31 
32 
33 
34 
35 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 



50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



P. P. 



43 



4 


3 


4 


5 





4. 


5 


7 


5. 


6 


A 


6. 


7 


i 


7- 


14 


3 


14. 


21 


5 


21. 


28 


6 


28. 


35 


8 


35. 



42_ 

2 
9 
6 
4 
1 
1 
2 
3 
4 



6 


4. 


7 


4. 


8 


5. 


9 


6. 


10 


7. 


20 


14. 


30 


21. 


40 


28. 


50 


35. 



43 

2 
9 
6 
3 








30 



6 

7 

8 

9 

10 

20 

30 

40 

50 



3 





2. 


3 


5 


3. 


4 





3. 


4 


5 


4. 


5 





4. 


10 





9. 


15 





14. 


20 





19. 


25 





24. 





39 


2 


6 


2-9 


2 • 


7 


3 


4 


3. 


8 


3 


8 


3. 


9 


4 


3 


4. 


10 


4 


8 


4. 


20 


9 


6 


9. 


30 


14 


5 


14. 


40 


19 


3 


19. 


50 


24 


1 


23. 



39 

9 
4 
9 
4 
9 
8 
7 
6 
6 



38 

2.8 

3.2 

37 

4.2 

4.6 

9-3 

14.0 

18-6 

23.3 



P. P. 



615 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 
48° 49° 



Lg. Vers. 



51965 
51994 
52022 
52050 
52079 



52107 
52135 
52164 
52192 
52220 



52249 
52277 
52305 
52333 
52362 



52390 
52418 
52446 
52474 
52503 



52531 
52559 
52587 
52615 
52643 



52671 
52699 
52727 
52756 
52784 



52812 
52840 
52888 
52898 
52924 



52952 
52980 
53008 
53036 
53084 



53092 
53120 
53147 
53175 
53203 



53231 
53259 
53287 
53315 
53343 



53370 
53398 
53426 
53454 
53482 



53509 
53537 
53565 
53593 
53620 



53648 



Lg. Vers. 



n 

28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 
28 
28 
28 
28 
28 

28 
28 
28 
28 
28 
28 
28 
28 
28 
28 

28 
28 
27 
28 
28 

28 
27 
28 
28 
28 

27 
28 
28 
27 
28 

27 
28 
27 
28 
27 
28 





.og.Exs. 


{ 


). 69414 
•69457 
.69499 
•69542 
.69584 


i 


). 69626 
.69669 
.69711 
•69753 
•69796 


€ 


•69838 
•69881 
•69923 
•69965 
•70008 


g 


• 70050 
-70092 
.70135 
•70177 
•70220 


9 


•70262 
.70304 
.70347 
.70389 
•70431 


9 


. 70474 
.70516 
•70558 
.70601 
.70643 


9 


.70685 
.70728 
.70770 
.70812 
•70854 


9 


.70897 
.70939 
•70981 
.71024 
•71066 


9 


.71108 
.71151 
.71193 
.71235 
•71278 


9 


.71320 
.71362 
•71404 
•71447 
.71489 


{ 


). 71531 
•71573 
•71616 
•71658 
•71700 


< 


3.71743 
•71785 
•71827 
•71869 
•71912 


\ 


9-71954 


\ 


Log.Exs. 



D 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 



Lg.Vers, 



9-53648 
53676 
53704 
53731 
53759 



53787 
53814 
53842 
53870 
53897 



53925 
53952 
53980 
54008 
54035 

54063 
54090 
54118 
54145 
54173 



54200 
54228 
54255 
54283 
54310 



54338 
54365 
54393 
54420 
54448 



54475 
54502 
54530 
54557 
54585 



54612 
54639 
54667 
54694 
54721 



54748 
54776 
54803 
54830 
54858 



5488 

54912 

54939 

54967 

5.4994 

55021 
55048 
55075 
55103 
55130 



55157 
55184 
55211 
55238 
55265 



9-55292 
Lg. Vers. 



n 

27 
28 
27 
27 

28 
27 
27 
28 
27 
27 
27 
28 
27 
27 

27 
27 
27 
27 
27 

27 

27 
27 
27 
27 

27 
27 
27 
27 
27 
27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 



9-71954 
71996 
72038 
72081 
72123 



Log.Exs. 



72165 
72207 
72250 
72292 
72334 



72376 
72419 
72461 
72503 
72545 



72587 
72630 
72672 
72714 
72756 



72799 
72841 
72883 
72925 
72967 



73010 
73052 
73094 
73136 
73178 



73221 
73263 
73305 
73347 
73389 



73431 
73474 
73516 
73558 
73600 



73642 
73685 
73727 
73769 
73811 



73858 
73895 
73938 
7398C 
7402 



74064 
74106 
74148 
74191 
74233 



74275 
74317 
74359 
74401 
74444 



74486 



Log.Exs. 



I) 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 

"77 



o 

1 

2 
3 

_4 
5" 
6 
7 
8 

_9 

10 

11 
12 
13 
11 
15 
16 
17 
18 

-ii 
30 

21 
22 
23 
_2_4 
25 
26 
27 
28 
2^ 

30 

31 
32 
33 

35 

36 

37 

38 

_39 

40 

41 
42 
43 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



P. P. 



6 

7 
8 
9 

10 
20 
30 
40 
50 



6 
7 
8 
9 

10 
20 
30 
40 
50 



43 



4 


2 


4 


9 


5 


6 


6 


4 


7 


1 


14 


1 


21 


2 


28 


3 


35 


4 



43 

4.2 

4-9 

5.6 

6.3 

7.0 

14.0 

21.0 

28.0 

35.0 



38 



2 


8 


2. 


3 


3 


3. 


3 


8 


3^ 


4 


3 


4^ 


4 


7 


4^ 


9 


5 


9^ 


14 


2 


14 • 


19 





18- 


23 


7 


23 • 



38 
8 
2 
7 
2 
6 
3 

6 
3 



37 



2^7 


2. 


3 


2 


Z- 


3 


6 


Z' 


4 


1 


4. 


4 


6 


4. 


9 


] 


9. 


13 


7 


13- 


18 


3 


18. 


22 


9 


22. 



37 

7 
I 
6 

5 

5 

5 



P. P. 



616 



TABLE Vlll.— LOGARITHMIC VEESED SINES AND EXTERNAL SECANTS. 
54^ 55"" 



Lg.Vers. 



55292 
55319 
55347 
55374 
55401 



55428 
55455 
55482 
55509 
55536 



55583 
55590 
55617 
55644 
55671 



55698 
55725 
55751 
55778 
55805 



55832 
55859 
55888 
55913 
55940 



55906 
55993 
56020 
56047 
56074 



56101 
56127 
56154 
56181 
56208 

56234 
56261 
56288 
56315 
56311 



56368 
56395 
56421 
56448 
56475 



56501 
56528 
56554 
56581 
56608 



56634 
56661 
56687 
56714 
56741 



56767 
56794 
56820 
56847 
56873 



9-56900 
Lg.Vers, 



2> 

27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
26 
27 
27 
27 
27 
26 
27 
27 

26 
27 
27 
26 
27 

27 

26 
27 
26 
27 
26 
27 
26 
27 
26 
26 
27 
26 
26 
27 

25 
26 
26 
27 
26 

26 
26 
26 
26 
27 

26 
26 
26 
26 
28 

26 



Log.Exs 



9-74486 
.74528 
•74570 
-74612 
-74654 



9-74696 
.74739 
.74781 
.74823 
.74865 



9 . 74907 
• 74949 
.74991 
.75033 
.75076 



9.75118 
.75160 
•75202 
.75244 
.75286 



9.75328 
.75370 
.75413 
.75455 
.75497 



9^ 75539 
•75581 
•75623 
•75660 
.75707 



9.75750 
.75792 
.75834 
.75876 
^9^ 

9.75960 
.76002 
•76044 
.76086 
.761!^R 



9.76171 
•76213 
.76255 
•76297 
.76339 



9.76381 
.76423 
.76465 
•76507 
.76549 



9.76592 
•76634 
•76676 
.76718 
-76760 



9.76802 
. 76341 
-768^5 
.76928 
•76970 



9-77012 
Log.Exs. 



n 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 



Lg. Vers. 



-56900 
.56926 
-56953 
.56979 
.57005 



9-57032 
.57058 
.57085 
.57111 
-57138 

9.57164 
•57190 
•57217 
•57243 
.57269 



9.57296 
.57322 
.57348 
.57375 
•57401 



9.57427 
.57454 
.57480 
.57506 
•57532 



9 • 57559 
.57585 
.57611 
.57637 
•57684 

9 •57890 
•57716 
. 57742 
.57768 
•57794 



9^57821 
.57847 
•57873 
.57899 



9 •57951 
•57977 
•58003 
•58029 
-58055 



9.58082 
•58108 
•58134 
.58160 
•58186 



-58212 
.58238 
•58264 
.58290 
-58316 



58342 
58367 
58393 
58419 
58445 



9-58471 
Lg. Vers. 



26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
28 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
25 
26 
26 
26 

26 



I) 



Log.Exs 



9-77012 

-77055 

-77097 

-77139 

77181 



9-77223 
-77265 
-77307 
-77349 
•77391 



9-77433 
.77475 
.77517 
.77560 
-77602 



. 77644 
•77686 
•77728 
•77770 
•77812 

9 •77854 
.77896 
.77938 
.77980 
•78022 



9.78064 
.78107 
.78149 
.78191 
•78233 



9-78275 
.78317 
.78359 
. 78401 
. 78443 



9 • 78485 
.78527 
•78589 
•78611 

-7865^ 



9.78696 
•78738 
•78780 
•78822 
-78864 

•78906 
•78948 
•78990 
.79032 
-79074 



9-79116 
•79158 
.79200 
.79242 
-79285 



9-79327 
-79369 
-79411 
-79453 
•79495 



9 79537 
Log.Exs. 



n 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 

IT 





1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
li 
15 
16 
17 
18 

11 
30 

21 
22 
23 
2i 
25 
26 
27 
28 
2i 
30 
31 
32 
33 
11 
35 
36 
37 
38 

il 
40 

41 
42 
43 
44 

45 
46 
47 
48 
Jt9 

50 

51 
52 
53 
_54 

55 
56 
57 
58 
-_59 
GO 



P.P. 



43 



6 

7 
8 

9 

10 
20 
30 
40 
50 



4 


2 


4- 


4 


9 


4. 


5 


6 


5. 


6 


4 


6. 


7 


1 


7- 


14 


1 


14- 


21 


2 


21. 


28 


3 


28. 


35 


4 


35- 



43 

2 
9 

3 










37 


37 


6 


2-7 


2.7 


7 


3-2 


3 


1 


8 


3^6 


3 


6 


9 


4-1 


4 





10 


4-6 


4 


5 


20 


9.1 


9 





30 


13-7 


13 


5 


40 


18^3 


18 





50 


22-9 


22 


5 



36 



2-6 


2- 


3 


1 


3- 


3 


5 


3- 


4 





3^ 


4 


4 


4:. 


8 


8 


8. 


13 


2 


13- 


17 


6 


17- 


22 


1 


21. 



36 

6 
(5 
4 
9 
3 
6 

3 
6 





35 


6 


2-5 


7 


3 





8 


3 


4 


9 


3 


g 


JO 


4 


2 


20 


8 


5 


30 


12 


7 


40 


17 





50 


21 


2 



P.P. 



617 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
53° 53° 



Lg. Vers, 



58471 
58497 
58523 
58549 
58575 



58601 
58626 
58652 
58678 
587C4 



58^30 
58755 
58781 
58807 
58833 



58859 
58884 
58910 
58936 
58962 



58987 
59013 
59039 
59064 
59090 



59116 
59141 
59167 
59193 
59218 



59244 
59270 
59295 
59321 
59346 



59372 
59397 
59423 
59449 
59474 



59500 
59525 
59551 
595'76 
59602 



59627 
59653 
59678 
59704 
59729 



59754 
59780 
59805 
59831 
59856 



59881 
59907 
59932 
59958 
59983 



960009 



Lg. Vers, 



26 
25 
26 
26 

26 
25 
26 
26 
25 

26 
25 
26 
26 
25 

26 
25 
26 
25 
26 

25 
25 
26 
25 
26 

25 
25 
26 
25 
25 

23 

26 
25 
25 
25 

25 
25 
26 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 



Log.Exs. 



79537 
79579 
79621 
79663 
79705 



79747 
79789 
79831 
79874 
79916 



79958 
80000 
80042 
80084 
80126 



80168 
80210 
80252 
.80294 
80S36 



80378 
80420 
80463 
80505 
80547 



80589 
80631 
80673 
80715 
80757 



80799 
80841 
80883 
80925 
80968 



81010 
81052 
81094 
81136 
81178 



81220 
81262 
81304 
81346 
81388 

81430 
81473 
81515 
81557 
81599 



81641 
81683 
81725 
81767 
81809 



81851 
81894 
81936 
81978 
82020 



82062 



1> Log.Exs 



1> 



42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 



Lg. Vers, 



60008 
60034 
60059 
60084 
60110 



60135 
60160 
60185 
60211 
60236 



60261 
60286 
60312 
60337 
60362 



60387 
60412 
60438 
60463 
60488 



60513 
60538 
60563 
60589 
60614 



60639 
60664 
60689 
60714 
60739 



60764 
60789 
60814 
6C839 
R0864 



60889 
6C914 
60939 
60964 
60989 



61014 
61039 
61064 
61089 
61114 



61139 
61164 
61189 
61214 
61239 



61264 
61289 
61313 
61338 
61363 



61388 
61413 
61438 
61462 
61487 



61512 



Lg. Vers, 



1> 

25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
24 
25 

25 
25 
24 
25 
25 

25 
24 
25 
24 
25 
25 



Log.Exs, 



82062 
82104 
82146 
82188 
82230 



82272 
82315 
82357 
82399 
82441 



82483 
82525 
82567 
82609 
82651 



82694 
82736 
82778 
82820 
82862 

82904 
82946 
82988 
83031 
83073 



83115 
83157 
83199 
83241 
83283 



83325 
83368 
83410 
83452 
83494 



83536 
83578 
83620 
83663 
83705 



83747 
83789 
83831 
83873 
83916 



83958 
84000 
84042 
84084 
84x26 



84168 
84211 
84253 
84295 
84337 

84379 
84422 
84464 
S4506 
84548 



9.84590 



2> 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 



i> Log.Exs. n 
~618 



10 

11 

12 
13 
14 



15 
]6 
17 
18 
19 

20 

21 

22 

23 

_24 

25 
26 
27 
28 
29 

30 

31 
32 
33 

-34 
35 
36 

F37 
38 
39 

40 

41 
42 
43 
J4 

45 
46 
47 
48 
j49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



P. P. 



6 

7 
8 
9 
10 
20 
30 
40 
50 



6 

7 
8 
9 
10 
20 
30 
40 
50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



43 



4 


2 


4 


4 


9 


4. 


5 


6 


5. 


6 


4 


6 


n 


1 


7- 


14 


1 


14 


21 


2 


21. 


28 


3 


28 


35 


4 


35. 



43 

2 
9 
6 
3 








36 



2 


6 


2 


3 





3. 


3 


4 


3 


3 


9 


3- 


4 


3 


4- 


8 


5 


8 


13 





12. 


17 


3 


17. 


21 


6 


21. 



35 

5 

4 
8 
2 
5 
7 

2 



35 



2 


5 


2. 


2 


9 


2. 


3 


3 


3. 


3 


7 


3. 


4 




4. 


8 


3 


8. 


12 


5 


12. 


16 


6 


16. 


20 


8 


20. 



34 

4 
8 
2 
7 
1 
I 
2 
3 
4 



P.P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
54° 55° 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
_9^ 

10 

11 
12 
13 
ii 
15 
16 
17 
18 
19 



20 

21 

22 

23 

21 

25 

26 

27 

28 

29_ 

30 

31 

32 

33 

31 

35 

36 

37 

33 

39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49^ 

50 

51 
52 
53 

54 

55 
56 
57 
58 
59 

60 



Lg. Vers 



61512 
61537 
61562 
61586 
6161] 



61635 
61661 
61685 
61710 
61733 



617G0 
61784 
61809 
61834 
81358 



61333 
61908 
61932 
61957 
61932 



62006 
62031 
62055 
62080 
62105 



62129 
62154 
62178 
62203 
62227 



62252 
62276 
62301 
62325 
62350 



62374 
62399 
62423 
62448 
82472 



62497 
62521 
62548 
62570 
62594 



62319 
62643 
62668 
62692 
62716 



62741 
62765 
62789 
62814 
62838 



82832 
62887 
S2911 
62935 
62960 



629 R4- 



Lg. Vers. D Log.Exs 



Log.Exs 



84590 
84632 
84675 
84717 
84759 



84801 
84843 
84886 
84928 

8497G 



8501 

85054 

85097 

85139 

85181 



85223 
85265 
85308 
85350 
85392 



85434 
85476 
85519 
85561 
85603 



85645 
85688 
85730 
85772 
85814 



85857 
85899 
85941 
85983 
88026 



86068 

86110 
86152 
86195 
83237 



88279 
83321 
86364 
88406 
86448 



86490 
86533 
86575 
86617 
86659 



88702 
88744 
86786 
86829 
86871 



88913 
86956 
88998 
87040 
87082 



87125 



D 



2> 



Lg. Vers, 



9-62984 
63008 
63032 
63057 
63081 



63105 
63129 
63154 
63178 
63202 



63226 
63250 
63274 
63209 
63323 



63347 
63371 
63395 
63419 
63443 

63468 
63492 
83516 
63540 
63584 



63598 
63612 
63836 
63680 
63684 



63708 
63732 
63756 
83780 

63804 



6382S 
63852 
63876 
63900 
63924 



63948 
63972 
63996 
64019 
64043 



84067 
84091 
64115 
64139 
64163 



64187 
84210 
64234 
64258 
64282 



64306 
64330 
84353 
84377 
64401 



64425 



Lg. Vers, 



7> 



24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 

24 

24 
24 
24 
23 
24 

24 
24 
24 
23 
24 

24 
23 
24 
24 
23 
24 
24 
23 
24 
23 
24 



JD 



Log.Exs. 



9-87125 
87187 
87209 
87252 
87294 



873S& 
87379 
87421 
87463 
87506 



87548 
87590 
87633 
87875 
87717 



87760 
87802 
87844 
87887 

87929 



87971 
88014 
88056 
88099 

88141 



88183 
88226 
88268 
88310 
88353 



88395 
88438 
88480 
88522 
88585 



88607 
88650 
88692 
88734 
88777 



88819 
88862 
88904 
88947 
S8989 



89031 
89074 
89116 
89159 
89201 



89244 
89286 
89329 
89371 
89414 



89456 
89499 
89541 
89583 
89828 



9. 89668 



Log.Exs, 



T> 



5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 

11 
30 

21 
22 
23 
24 

25 
26 
27 
28 
29, 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
48 
47 
48 
49 

50 

51 
52 
53 
54 

55 
58 
57 
58 
59, 
60 



P.P. 



6 

7 

8 

9 

10 

20 
30i21 
40i28 
50135 



4^ 
4.2 
4 
5 
6 
7 
14 



42 

4.2 

4.9 

5.6 

6.3 

7-0 

14.0 

21.0 

28.0 

35.0 





25 


24 


6 


2.5 


2.4 


7 


2.9 


2.8 


8 


3.3 


3.2 


9 


3-7 


3.7 


10 


4-1 


4-1 


20 


8-3 


81 


30 


12-5 


12-2 


40 


18-6 


16.3 


50 


20.8 


20.4 



24 


23 


2-4 


2.3 


2.8 


2 


7 


3.2 


3 


1 


3-6 


3 


5 


4.0 


3 


9 


8-0 


7 


8 


12.0 


11 


7 


16.0 


15 


5 


20.0 


19 


6 



P.P. 



619 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 

56° 57° 



O 

1 
2 
3 

5 
6 
7 
8 
_9^ 

10 

11 

12 

13 

li 

15 

16 

17 

18 

ii 

20 

21 

22 

23 

21 

25 

26 

27 

28 

2i 

30 

31 

32 

33 

34 

35 
36 
37 
38 

SI 
40 

41 
42 
43 
4£ 

45 
46 
47 
48 
4i 

50 

51 
52 
53 
54 

55 

56 

57 

58 

59_ 

60 



Lff.Vers. 



64425 
64448 
64472 
64496 
64520 



64543 
64567 
64591 
64614 
64638 

64662 
64685 
64709 
64733 
64756 



I> 



64780 
64804 
64827 
64851 
64875 



64898 
64922 
64945 
64969 
64992 



65016 
6504Q 
65063 
65087 
65110 



65134 
65157 
65181 
65204 
65228 



65251 
65275 
65298 
65321 
653^5 



65368 
65392 
65415 
65439 
65462 

85485 
65509 
65532 
65556 
65579 



65602 
65626 
65649 
65672 
65696 



65719 
65742 
65765 
65789 
65812 



"^ 65835 



Lg. Vers. 



23 
24 
23 
24 

23 
24 
23 
23 
24 

23 
23 
24 
23 
23 

24 
23 
23 
23 
24 

23 
23 
23 
23 
23 

24 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 



Log.Exs. 



9.89668 
.89711 
•89753 
•89796 
•89838 



9 •89881 
•89923 
•89966 
.90008 
•90051 



9-90094 
•90136 
•90179 
•90221 
•90264 



2> 



9.90306 
•90349 
•90391 
.90434 
•90476 



9.90519 
•90561 
.90604 
.90647 
.90689 



9.90732 
90774 
90817 
90860 
90902 



9-90945 
-90987 
•91030 
•91073 
•91115 



9^91158 
•91200 
•91243 
•91286 
.913?»R 



9^91371 
91414 
91456 
91499 
91541 



9-91584 
•91627 
-91669 
•91712 
•91755 



9-91797 
•91840 
-91883 
•9192B 
•91968 



9^92011 
.92054 
.92096 
.92139 
•92182 



9.9222A 



Log.Exs. 



42 

42 
42 
42 

42 
42 
42 
42 
42 

43 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
43 
42 
42 

42 
42 
42 
43 
42 

42 
42 
42 
43 
42 

42 
42 
42 
43 
42 

42 
43 
42 
42 
42 

43 
42 
42 
43 
42 

42 
43 
42 
43 
42 

42 
43 
42 
43 
42 

42 



Lg, Vers. 



•65835 
•65859 
•65882 
-65905 
•65928 



65952 
65975 
65998 
66021 
66044 



66068 
66091 
66114 
66137 
66160 



66183 
66207 
66230 
66253 
66276 



66299 
66S22 
66345 
66368 
66391 



9- 



66415 
66438 
66461 
66484 
66507 



68530 
66553 
66576 
66599 
66622 



66645 
66668 
66691 
66714 
66737 



J> 



66760 
66783 
66805 
66828 
66851 



66874 
66897 
66920 
66943 
6696P 



66989 
67012 
67034 
67057 
67080 



67103 
67126 
67149 
67171 
67194 



9-67217 



Z> Lg.Vers, 



23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
22 
23 

23 
23 
22 
23 
23 

22 
23 
23 
22 
23 
22 



Log.Exs. 



92224 
92267 
92310 
92353 
92395 



2> 



92438 
92481 
92524 
92566 
92609 



92652 
92695 
92737 
92780 
92823 



92866 
92909 
92951 
92994 
93037 



93080 
93123 
93165 
93208 
93251 



98294 
93337 
93380 
93422 
93465 



93508 
93551 
93594 
93637 
93680 

93722 
93765 
93808 
93851 
93894 



93937 
93980 
94023 
94066 
94109 



94151 
94194 
94237 
94280 
94328 



94366 
94409 
94452 
94495 
94538 



94581 
S4624 
94667 
94710 
94753 



94796 



43 
42 
43 
42 

43 
42 
43 
42 
43 

42 
43 
42 
43 
42 

43 
43 
42 
43 
42 

43 
43 
42 
43 
43 
42 
43 
43 
42 
43 

43 
42 
43 
43 
43 
42 
43 
43 
43 
42 

43 
43 
43 
43 
43 

42 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 



O 

1 
2 
3 
4 

5 
6 
7 
8 
_9 

10 

11 

12 
13 
-14 

15 
16 
17 
18 

ii 
30 

21 
22 
23 
24 

25 
26 
27 
28 
29_ 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
-44 

45 
46 
47 
48 
49 



50 

51 
52 
53 
54 



1> Log.Exs. Z> 
620 



60 



P. P. 



43 



4 


3 


4. 


5 





4.. 


5 


7 


5- 


6 


4 


6- 


7 


1 


7- 


14 


3 


14- 


21 


5 


21. 


28 




28- 


35 


8 


35- 



43 

2 
9 
6 
4 
1 
I 
2 
3 
4 



34 23 



6 
7 
8 
9 

10 
20 
30 
40 
50 



2-4 


2 


2-8 


2. 


3-2 


3- 


3-6 


3- 


4-0 


3. 


8.0 


7- 


12-0 


11- 


16-0 


15. 


20-0 


19- 





23 


6 


2^3| 


7 


2 


• 7 


8 


3 





9 


3 


4 


10 


3 


8 


20 


7 


6 


30 


11 


5 


40 


15 


3 


50 


19 


1 



33_ 

2-2 

2-6 

3-0 

3-4 

3-7 

7-5 

11-2 

15-0 

18-7 



P.P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

58° 59° 



Lg. Vers. 



9-67217 
67240 
67263 
67285 
67308 



67331 
67354 
67376 
67399 

67422 



67445 
6746? 
67490 
67513 
67535 



67558 
67581 
67603 
67626 
67^4P 



67671 
67694 
67717 
67739 
67762 



67784 
67807 
67830 
67852 
67875 



67897 
67920 
67942 
67965 
67987 



68010 
68032 
68055 
68077 
68100 



68122 
68145 
68167 
68190 
68212 



68235 
68257 
68280 
68302 
68324 



68347 
68369 
68392 
68414 
68436 



68459 
68481 
68503 
68526 
68548 



68571 



Lg. Vers, 



J> 



n 



Loer.Exs. 



9.94796 
.94839 
•94832 
.94925 
.94968 

9.95011 
.95054 
.95097 
.95140 
.95183 



9-95226 
.95269 
.95313 
.95356 
•95399 



9.95442 
.95485 
.95528 
.95571 
.9561^ 



9-95657 
.95700 
.95744 
.95787 
-95830 



9-95873 
.95916 
.95959 
.96002 
-96046 



9-98089 
.96132 
.96175 
.96218 
-96281 



9-96305 
.96348 
.98391 
.96434 

.9?iA7« 



9-96521 
.96564 
.96807 
-98850 
-96394 



9-93737 
.98780 
.96824 
.98387 
• 95^)1 n 



9-96953 
-96997 
-97040 
.97083 
-97127 



9-97170 

•97213 

.97257 

•97300 

_^97343 

9. 9 73 8 7 



Log. Exs 



J) 

43 
43 
43 
43 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 



Ls^. Vers. 



9^68571 
68593 
38615 
68637 
68660 



63382 
68704 
68727 
68749 
68771 



68793 
68816 
68838 
68830 
68882 



63905 
68927 
68949 
68971 
6899.^ 



690] 6 
69038 
69060 
69082 
69104 



69126 
69149 
69171 
69193 
69215 



69237 
69259 
69281 
69303 
69325 



69347 
69369 
69392 
69414 
69436 



69458 
69480 
69502 
69524 
69546 



69568 
69590 
69612 
69634 
69656 



69678 
69700 
69721 
69743 
69765 



69787 
69809 
69831 
69853 
69875 



1i 



q.fiP897 
jLg. Vers 



jy 



22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
2l 
22 
22 

22 
22 
22 
21 
22 

22 



Log. Exs. 



97387 
97430 
97473 
97517 
97560 



97603 
97647 
97690 
97734 
97777 



97820 
97864 
97907 
97951 
97994 



98038 
98081 
98125 
98168 
98211 



98255 
98298 
98342 
98385 
98429 



98472 
98516 
98559 
98603 
98647 



98690 
98734 
98777 
98821 
98864 



98008 
98952 
98995 
99039 
99082 



99126 
99170 
99213 
99257 
99300 



99344 
99388 
99431 
99475 
99519 



99562 
99606 
99650 
99694 
99737 

99781 
99825 
99868 
99912 
99956 



in.oonno 



Log. Ex5 



J> 

43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
44 
43 

43 
43 
43 
43 
43 

43 
4| 
43 
43 
43 

43 
44 
43 
43 
43 

44 
43 
43 
44 
43 

43 
44 
43 
44 
43 
44 
43 
43 
44 
43 

44 



2> 





1 

2 

3 

_4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
il 
15 
16 
17 
18 
19 

20 

21 
22 
23 
2k 
25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 

44 

45 
46 
47 
48 
Jt9 

50 

51 
52 
53 
54 

55 
56 
57 
58 
_59 
60 



P. P. 





44 


43 


6 


4.4 


4^3 


7 


5^1 


5.1 


8 


5.8 


5-8 


9 


6^6 


65 


10 


7^3 


7.2 


20 


14.6 


14.5 


30 


22.0 


21-7 


40 


29.3 


29-0 


50 


36-6 


36.2 





43 


6 


4.3 


7 


5.0 


8 


5.^ 


9 


6.^: 


10 


7. 


20 


14. c 


30 


21-5 


40 


28-6 


5C 


35.8 



33 



6 
7 
8 
9 

10 
20 
30 
40 
50 



2 


3 


2- 


2 




2. 


3 





3. 


3 


4 


3- 


3 


8 


3. 


7 


5 


7. 


11 


5 


11. 


15 


3 


15. 


19 


1 


18. 



23 
2 
6 

4 
7 
5 
2 

7 



23 

2.2 



21. 

2-1 
2.5 
2.8 
3.2 
3.6 

oiiol? 

614-3 
3J7.9 



P. P. 



G21 



TABLE \ail.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
60° 61° 



Lg.Vers. 



69897 
69919 
69940 
69962 
69984 



70006 
70028 
70050 
70072 
70093 



70115 
70137 
70159 
70181 
70202 



70224 
70246 
70268 
70289 
70311 



O 

1 
2 
3 

5 
6 
7 
8 
_9^ 

10 

11 
12 
13 
li 
15 
16 
17 
18 

11 
30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 9.70766 



70333 
70355 
70376 
70398 
70420 



70441 
70463 
70485 
70507 
70528 



9-70550 
•70572 
.70593 
.70615 
.70636 



9-70658 
.70680 
.70701 
.70723 
-70745 



41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



.70788 
.70809 
.70831 
-70852 



9-70874 
.70896 
.70917 
-70939 
-70960 



9-70982 
.71003 
-71025 
.71046 
-71068 



9-71089 
.71111 
.71132 
.71154 
-71175 



2> 



9. 71197 



Lg. Vers. 



22 
21 
22 
22 

22 
21 
22 
22 
21 

22 
21 
22 
22 
21 

22 
21 
22 
21 
22 

2l 
22 
21 
22 
21 

2l 
22 
21 
22 
21 

2l 
22 
21 
21 
21 

22 
21 
21 
21 
22 

2l 
21 
21 
2l 
21 

22 
21 
21 
21 
21 

21 
21 
21 
2l 
2l 

21 
2T 
2T 
2l 
21 

2l 



Log. Exs. 



10.00000 
.00044 
.00087 
.00131 
-00175 



10-00219 
.00262 
.00306 
.00350 
-00394 



10-00438 
.00482 
.00525 
.00569 
-00613 



10.00657 
.00701 
.00745 
.00789 
-00833 



10-00876 
.00920 
.00964 
.01008 
-01052 



10.01096 
.01140 
.01184 
.01228 
.01272 



10.01316 
.01360 
.01404 
.01448 
.01492 



10.01536 
.01580 
.01624 
.01668 
.01712 



10.01756 
.01800 
.01844 
.01889 
.01933 



10.01977 
.02021 
.02065 
.02109 
-02153 



Z> 



10-02197 
.02242 
.02286 
.02330 
-02374 



10-02418 
.02463 
.02507 
.02551 
-02595 



10.02639 



44 
43 
44 
43 

44 
43 
44 
44 
43 

44 
44 
43 
44 
44 

44 
43 
44 
44 
44 

43 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 



Log. Exs, 



Lg. Vers. 



71197 
7121C 
712^9 
71261 
71282 



71304 
71325 
71346 
71368 
71389 



71411 
71432 
71453 
71475 
71496 



71517 
71539 
71560 
71581 
71603 



71624 
71645 
71667 
71688 
71709 

71730 
71752 
71773 
71794 
71815 



71837 
71858 
71879 
71900 
71922 



71943 
71964 
71985 
72006 
72028 



72049 
72070 
72091 
72112 
7213^ 



72154 
72176 
72197 
72218 
72239 



72260 
72281 
72302 
72323 
72344 

72365 
72386 
72408 
72429 
72450 



72471 



-D Lg.Vers 



10- 



10 



10 



10 



10 



1> 

21 
21 
21 
21 

21 
21 
21 
21 
21 

2l 10 
21 ^" 

21 

21 

21 

21 
21 
21 
21 
21 
21 
21 
21 
21 
21 

21 
21 
21 
21 
21 
2l 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
2l 
21 
21 
21 

2l 
21 
21 
21 
21 

21 
21 
2l 
21 
21 
21 



Log. Exs. 



02639 
02684 
02728 
02772 
02816 



2> 



02861 
02905 
02949 
02994 
03038 



03082 
03127 
03171 
03215 
03260 



03304 
03348 
03393 
03437 
03481 



03526 
03570 
03615 
03659 
03704 



03748 
03793 
03837 
03881 
03926 



10 



03970 
04015 
04059 
04104 
04149 



10 



04193 
04238 
04282 
04327 
04371 



10 



0441 6 
04461 
04505 
04550 
04594 



10 



04639 
04684 
04728 
04773 
P4818 



10 



04862 
04907 
0495? 
04996 
05041 



10 



05086 
.05131 
.05175 
05220 
05265 



10.05310 



-D Log. Exs. 
622 



44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
45 

44 
44 
44 
44 
44 

44 
45 
44 
44 
44 

45 
44 
44 
44 
45 

44 
45 
44 
44 
45 

44 
45 
44 
45 
44 

45 



5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 

ii 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 

_34 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
-49 
50 
51 
52 
53 
-54 
55 
56 
57 
58 
59 

GO 



P.P. 



45 44 



6 

7 
8 
9 
10 
20 
3C 
4G 
50 



4- 


5 


4- 


4 


5 


2 


5 


2 , 


6 





5 


9 ' 


6 


7 


6 


7 


7 


5 


7 


4 


15 





14 


8 


22 


5 


22 


2 


30 





29 


6 


37 


5 


37 


1 





44 


43. 


6 


4 4 


4-3 


7 


5 


1 


5 


1 


8 


5 


8 


5 


8 


9 


6 


6 


6 


5 


10 


7 


3 


7 


2 


20 


14 


6 


14 


5 i 


30 


22 





21 


7 ' 


40 


29 


3 


29 





50 


36 


6 


36 


-2 



23 



6 

7 

8 

9 

10 

20 

30 

40 

50 



2 


2 


2 


5 


2 


9 


3 


3 


3 


6 


7 


3 


11 





14 


6 


18 


3 



31 

2-1 

2-5 

2-8 

3-2 

3-6 

71 

10-7 

14-3 

17.9 



31 

2.1 

2.4 

2-8 

3-1 

3-5 

7-0 

10.5 

14-0 

17.5 



P.P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
63° 63° 



Lg. Vers, 



72471 
72492 
72513 
72534 
72555 



72576 
72597 
72618 
72639 
72660 



72681 
72701 
72722 
72743 
72764 



72785 
72806 
72827 
72848 
72869 



72890 
72911 
72931 
72952 
72973 



72994 
73015 
73036 
73057 
73077 



73098 
73119 
73140 
73161 
73181 



73202 
73223 
73244 
73265 
73285 



73306 
73327 
73348 
73368 
73389 



73410 
7343Q 
73451 
73472 
73493 



73513 
73534 
73555 
73575 
73596 



73617 
73637 
73658 
73679 
73699 



9-73720 
Lg. Vers. 



I> 



21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
20 
21 
21 
21 
2] 
21 
21 
20 
21 

21 
21 
20 
21 
21 

21 
20 
21 
21 
20 

21 
21 
20 
21 
20 

21 
21 
20 
21 
20 

21 
20 
21 
20 
21 

ii 
i^ 

21 

20 
20 
21 
20 
20 

21 
20 
20 
21 
20 
20 



Log, Exs, 



10.05310 
.05354 
.05399 
.05444 
.05489 



10.05534 
.05579 
.05623 
.05668 
.05713 



10.05758 
.05803 
.05848 
.05893 
.05938 



10.05983 
.06028 
.06072 
.06117 
.06162 



10.06207 
.06252 
.06297 
.06342 
.06387 



10.06432 
.06477 
.06522 
.06568 
.06613 



10.06658 
.06703 
.06748 
.06793 
.06838 



10.06883 
.06928 
.06974 
.07019 
.07084 



10.07109 
.07154 
.07200 
.07245 
.07290 



10.07335 
.07380 
.07426 
.07471 
.07516 



10.07562 
.07607 
.07652 
.07697 
.07743 



10.07788 
.07834 
.07879 
.07924 
.07970 



10-08015 
Log, Exs, 



2> 



44 
45 
45 
44 

45 
45 
44 
45 
45 

45 
44 
45 
45 
45 
45 
45 
44 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 



Lg. Vers 



9-73720 
.73740 
•73761 
.73782 
-73802 



9.73823 
.73843 
•73864 
.73884 
•73905 



-73926 
.73946 
.73967 
.73987 
•74008 



9-74028 
• 74049 
.74069 
.74090 
•74110 



9.74131 
.74151 
.74172 
.74192 
.74213 



9.74233 
.74254 
.74274 
.74294 
.74315 



9-74335 
.74356 
.74376 
.74396 
•74417 



9 . 74437 
.74458 
.74478 
.74498 
-74519 



9-74539 
•74559 
.74580 
.74600 
.74620 



9 . 74641 
.74661 
.74681 
.74702 
.74722 



9 . 74742 
.74762 
.74783 
.74803 
-74823 



9 - 74844 
.74864 
. 7488^ 
.7490^ 
-74924 



9-74945 
Lg. Vers, 



D 



20 
20 
21 
20 

20 
20 
20 
20 
21 
20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 



Log, Exs. 



10-08015 
.08061 
.08106 
.08151 
.08197 



10.08242 
.08288 
.08333 
.08379 
-08424 



10-08470 
.08515 
.08561 
.08606 
•08652 



10.08697 
.08743 
.08789 
.08834 
.08880 



10.08926 
.08971 
.09017 
.09062 
.09108 



10.09154 
.09200 
.09245 
.09291 
.09337 



10.09382 
.09428 
.09474 
.09520 
-09566 



10.09611 
.09657 
.09703 
.09749 
-09795 



10-09841 
.09886 
.09932 
.09978 
.10024 



10.10070 
.10116 
.10162 
•10208 
-10254 



10-10300 
•10346 
•10392 
•10438 
-10484 



10-10530 
•10576 
.10622 
•10668 
.10714 



10-10760 
Log. Exs. 



2> 





1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
JA 
15 
16 
17 
18 
19 

20 

21 

22 

23 

-24 

25 
26 
27 
28 
-2i 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 

M. 

60 



P. P. 





46 


6 


4.6 


7 


5.4 


8 


6.2 


9 


7.0 


10 


7.7 


20 


15.5 


30 


23.2 


40 


31.0 


50 


38.71 





45 


6 


4.5 


7 


5.3 


8 


6.0 


9 


6.8 


10 


7.6 


20 


15.1 


30 


22.7 


40 


30.3 


50 


37.9 



46 

4.6 

5.3 

6.1 

6.9 

7-6 

15.3 

23.0 

30.6 



45 

4-5 

5-2 

6-0 

6^7 

75 

15.0 

22-5 

30-0 

37.5 



6 

7 

8 

9 

10 

20 

30 

40 

50 



44 

4-4 

52 

5.9 

6.7 

7-4 

14-8 

22-2 

29.6 

37.1 



31 

2-1 

2.4 

2-8 

3-1 

3^5 

7-0 

10.5 

14^0 

17.5 



30_ 

2-0 

2-4 

2-7 

3-1 

3-4 

68 

10.2 

13.6 

17.1 



6 


2. 


7 


2. 


8 


2- 


9 


3^ 


10 


3. 


20 


6 


30 


10 


40 


is- 


50 


le. 



30 


3 
6 

3 
6 

3 
6 



P. P. 



623 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
64° 65° 



Lg.Vers. 



45 9 

46 

47 

48 

49 



74945 
74965 
74985 
75005 
75026 



75046 
75066 
75086 
75106 
75126 



75147 
75167 
75187 
75207 
75227 



75247 
75267 
75287 
75308 
75328 



75348 
75368 
75388 
75408 
75428 



75448 
75468 
75488 
75508 
75528 



75548 
75568 
75588 
75608 
75628 



75648 
75668 
75688 
75708 
75728 



75748 
75768 
75788 
75808 
75828 



75848 
75868 
75888 
75908 
75928 



75947 
75967 
75987 
76007 
76027 



76047 
76067 
76087 
76106 
76126 



9-76146 



I) 

20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
19 

20 
20 
20 
20 
20 

19 
20 
20 
20 
19 

20 
20 
20 
19 
20 

20 



Log.Exs. 



10 



10760 
10807 
10853 
10899 
10945 



10 



10991 
11037 
11084 
11130 
11176 



10 



11222 
11269 
11315 
11361 
11407 



10 



11454 
11500 
11546 
11593 
11639 



10 



11685 
11732 
11778 
11825 
11871 



10 



11917 
11964 
12010 
12057 
12103 



10 



12150 
12196 
12243 
12289 
12336 



10 



12383 
12429 
12476 
12522 
12569 



10 



12616 
12662 
12709 
12756 
12802 



10 



12849 
12896 
12942 
12989 
13036 



10 



Lg, Vers. 1> Log.Exs 



13083 
13130 
.13176 
.13223 
.13270 



10 



13317 
13364 
13411 
13457 
13504 



10.13551 



46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

47 
46 
46 
46 
46 

47 
46 
46 
47 
46 

46 
47 
4C 
47 
46 

47 
47 
46 
47 
46 

47 
47 
47 
46 
47 

47 



Lg. Vers. 



9.76146 
.76166 
.76186 
•76206 
.76225 



9.76245 
76265 
76285 
76304 
76324 



9-76344 

76364 

.76384 

.76403 

•76423 



9.76443 
•76463 
•76482 
•76502 
•76522 



9.76541 
76561 
76581 
76600 
76620 



9.76640 
•76659 
•76679 
•76699 
.76718 



9.76738 
.76758 
•76777 
.76797 
.76817 



.76836 
.76856 
-76875 
.76895 
■76915 



J> 



9-76934 
•76954 
•76973 
.76993 
•77012 



9-77032 
.77052 
-77071 
-77091 
•77110 



2> 



9-77130 
•77149 
.77169 
.77188 
•77208 



9-77227 
-77247 
•77266 
•77286 
-77305 



9^77325 



Lg. Vers 



19 
20 
20 
19 
20 
19 
20 
19 
20 

20 
19 
20 
19 
20 

19 
20 
19 
19 
20 

19 
20 
19 
19 
20 

19 
19 
20 
19 
19 

20 
19 
19 
19 
20 

19 
19 
19 
20 
19 

19 
19 
19 

19 
19 

20 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 



Log.Exs. 



10.13551 
•13598 
.13645 
•13692 
-13739 



10.13786 
-13833 
-13880 
-13927 
•13974 



10-14021 
•14068 
•14115 
-14162 
•14210 



10^14257 
•14304 
.14351 
.14398 
• 14445 



10-14493 
• 14540 
-14587 
.14634 
-14682 



10-14729 
.14776 
•14823 
.14871 
•14918 



10.14965 
.15013 
.15060 
.15108 
.15155 



10.15202 
.15250 
.15297 
.15345 
.15392 



D 



10.15440 
.15487 
.15535 
.15582 
•15630 



10^15678 
•15725 
•15773 
•15820 
.15868 



10.15916 
.15963 
•16011 
•16059 
.16106 



10-16154 
-16202 
-16250 
.16298 
.16345 



10-16393 



Log.Exs. 



47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
4^ 
47 
47 
47 
47 
47 
47 
47 
47 

48 
42 

47 
47 
48 

47 
47 
48 
47 
47 

48 
47 
48 
48 
47 
48 



10 

11 

12 

13 

-14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

50 

51 
52 
53 

55 
56 
57 
58 

60 



P.P. 





4 


3 


47_ 


6 


4 8 


4-7 


7 


5 


6 


5 


5 


8 


6 


4 


6 


3 


9 


7 


2 


7 


1 


10 


8 





7 


9 


20 


16 





15 


8 


30 


24 





23 


7 


40 


32 





31 


6 


50 


40 





39 


6 



47 

4-7 

5-5 

6-2 

7-0 

7-8 
15-6 
23-5:23 
31-3131 
39-1138 



46 

4-6 



46 

4.6 



30 23 
40 30 
5038 



20_ 

2-0 



17.1 



20 

2-0 



6 


1- 


7 


2. 


8 


2- 


9 


2. 


10 


3- 


20 


6- 


30 


9. 


40 


13- 


50 


16 • 



19 

-9 
3 
6 

• 9 
-2 
•5 

7 

•Q 

• 2 



P.P. 



624 



TABLE VIII.—LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 
66° 67° 



O 

1 
2 
3 

j4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 

15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Lg.Vers 



77325 
77344 
77363 
77383 
77402 



77422 
77441 
77461 
77480 
77499 



77519 
77538 
77557 
77577 
77596 



77616 
77635 
77654 
77674 
77693 



77712 
77732 
77751 
77770 
77790 



77809 
77828 
77847 
77867 
77886 



77905 
77925 
77944 
77963 
77982 



78002 
78021 
78040 
78059 
78078 



78098 
78117 
78136 
78155 
78174 



78194 
78213 
78232 
78251 
78270 



78289 
78309 
78328 
78347 
78366 



78385 
78404 
78423 
78442 
78462 



9-78481 



Lg. Vers, 



19 

19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
1? 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 
19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 



Log.Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



16393 
16441 
16489 
16537 
16585 



16630 
16680 
16728 
16776 
16824 



16872 
16920 
16968 
17016 
17064 



17112 
17160 
17209 
17257 
17305 



17353 
17401 
17449 
17498 
17546 



17594 
17642 
17690 
17739 
17787 



17835 
17884 
17932 
17980 
18029 



18077 
18126 
18174 
18222 
18271 



18319 
18368 
18416 
18465 
18514 



18562 
18611 
18659 
18708 
18757 

18805 
18854 
18903 
18951 
19000 



19049 
19098 
19146 
19195 
19244 



10-19293 

Log.Exs. 



n 

48 
47 
48 
48 

48 
47 
48 
48 
48 

48 
48 
48 
48 
48 
48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
49 
48 

48 
48 
48 
48 
49 

48 
48 
49 
48 
49 

48 
49 
48 
49 
49 

48 



Lg. Vers 



9.78481 
.78500 
•78519 
.78538 
■78557 



9-78576 
.78595 
•78614 
•78633 

^78652 

9-78671 
•78690 
.78709 
.78728 
-78747 



9-78766 
.78785 
.78804 
.78823 
-78842 



-78861 
•78880 
.78899 
•78918 
•78937 

9-78956 
•78975 
.78994 
.79013 
-79032 



9-79051 
•79069 
•79088 
•79107 
•79126 



-79145 
•79164 
.79183 
•79202 
-79220 



9-79239 
.79258 
.79277 
.79296 
-79315 

9-79333 
.79352 
.79371 
.79390 
.79409 



9-79427 
. 79446 
.79465 
.79484 
-79503 



9-79521 
-79540 
•79559 
•79578 
.79596 



9-79615 



Lg. Vers. 



Log.Exs, 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



19293 
19342 
19391 
19439 
19488 



19537 
19586 
19635 
19684 
19733 



19782 
19831 
19880 
19929 
19979 



20028 
20077 
20126 
20175 
20224 



20273 
20323 
20372 
20421 
20470 



20520 
20569 
20618 
20668 
20717 



20767 
20816 
20865 
20915 
20964 



21014 
21063 
21113 
21162 
21212 



21262 
21311 
21361 
21410 
21460 



21510 
21560 
21609 
21659 
21709 



21759 
21808 
21858 
21908 
21958 



22008 
22058 
22108 
22158 
22208 



10-22258 
Log.Exs. 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 

12 

13 

J4 

15 
16 
17 
18 
ii 
20 
21 
22 
23 
2^ 

25 
26 
27 
28 
2^ 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39, 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



P. P. 



50 



6 

7 
8 
9 
10 
20 
30 
40 
50 



5 





5 


8 


6 


6 


7 


5 


8 


3 


16 


6 


25 





33 


3 


41 


6 





49 


6 


4-91 


7 


5 


7 


8 


6 


5 


9 


7 


3 


10 


8 


\ 


20 


16 


3 


30 


24 


5 


40 


32 


3 


50 


40 


8 





48 


6 


4-8 


7 


5^6 


8 


6-4 


9 


7-2 


10 


8-0 


20 


16-0 


30 


24-0 


40 


32^0 


50 


40.0 



6 


1-9 


7 


2-3 


8 


2-6 


9 


2.9 


10 


3.2 


20 


6.5 


30 


9.7 


40 


13^0 


50 


16.2 



49 

4^9 

5-8 

6^6 

7-4 

8^2 

16-5 

24^7 

330 

41^2 



48 

4^8 

5.§ 

6-4 

7-3 

81 

16.1 

24.2 

32.3 

40.4 



47 

4.7 

5.5 

6-3 

7-1 

7.9 

15-8 

23-7 

31-6 

39-6 



19 

1-9 
2.2 
2.5 
2.8 
3.1 
6.3 
9.5 
12.6 
15.8 



18 

l-I 
2.1 
2.5 
2.8 
31 
6.1 
9.2 
12^3 
15.4 



P. P. 



625 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



68' 



69' 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19. 

20 

21 
22 
23 

24 

25 
26 
27 
28 
29, 
30 
31 
32 
33 
M 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Lg, Vers. 



79615 
79634 
79653 
79671 
79690 



79709 
79727 
79746 
79765 
79783 



79802 
79821 
79839 
79858 
79877 



79895 
79914 
79933 
79951 
79970 



79988 
80007 
80026 
80044 
80063 



80081 
80100 
80119 
80137 
80156 



80174 
80193 
80211 
80230 
80248 



80267 
80286 
80304 
80323 
80341 



80360 
80378 
80397 
80415 
80434 



80452 
80470 
80489 
80507 
80526 



80544 
80563 
80587 
80600 
80618 



80636 
80655 
80673 
80692 
80710 



80728 



Lg. Vers. 



JD 

18 

19 
18 
18 

19 
18 
1? 
18 
18 

19 
18 
18 
19 
18 

18 
18 
19 
18 
18 

18 
19 
18 
18 
18 

18 
19 
18 
18 
18 

18 
18 
18 
18 
18 

19 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 



Log.Exs. 



10.22258 
.22308 
.22358 
.22408 
.22458 



10-22508 
.22558 
.22608 
.22658 
.22708 



10-22759 
.22809 
.22859 
.22909 
.22960 



10.23010 
.23060 
.23110 
.23161 
.23211 



10.23262 
.23312 
.23362 
-23413 
-23463 



10.23514 
.23564 
.23615 
.23666 
.23716 



10.23767 
.23817 
.23868 
•23919 
-23969 



10-24020 
-24071 
.24122 
.24172 
-24223 



10.24274 
.24325 
.24376 
.24427 
.24478 



10-24529 
.24580 
.24631 
.24682 
-24733 



10-24784 
.24835 
.24886 
.24937 
.24988 



10.25038 
.25090 
.25142 
.25193 
-25244 



10. 25295 



Log.Exs. 



Lg. Vers. 



9.80728 
-80747 
-80765 
.80783 
.80802 



9-80820 
-80839 
.80857 
.80875 
.80894 



9-80912 
.80930 
.80949 
.80967 
-80985 



9-81003 
.81022 
.81040 
.81058 
-81077 



9-81095 
.81113 
.81131 
.81150 
.81168 



9-81186 
. 81204 
.81223 
.81241 
-81259 



9.81277 
81295 
81314 
81332 
81350 



9.81368 
.81386 
.81405 
-81423 
.81441 



9.81459 
-81477 
.81495 
.81513 
-81532 



9.81550 
.81568 
.81586 
.81604 
.81622 



9.81640 
-81658 
-81676 
-81695 
.81713 



9-81731 
.81749 
.81767 
.81785 
.81803 



9.81821 



Lg.Vers, 



jy 



18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 
18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 



Log.Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



25295 
25347 
25398 
25449 
25501 



25552 
25604 
25655 
25707 
25758 



2581C 
25861 
25913 
25964 
26016 



26067 
26119 
26171 
26222 
26274 



26326 
26378 
26429 
26481 
26533 
26585 
26637 
26689 
26741 
26793 



26845 
26897 
26949 
27001 
27053 



27105 
27157 
27209 
27261 
27314 



27366 
27418 
27470 
27523 
27575 



22627 
27680 
27732 
27785 
27837 



27890 
27942 
27995 
28047 
28100 



28152 
28205 
28258 
28310 
28363 



28416 



T> Log.Exs. 
626 





1 

2 
3 
4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 



25 
26 
27 
28 
2^ 

30 

31 
32 
33 
M. 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 

60 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50, 



6 

7 

8 

9 

10 

20 

30 

40 

50 



53 

5-3 



53_ 

5-2 



53 

5 



51 

5.1 



51 



5 


1 


5. 


5 


9 


5- 


6 


8 


6. 


7 


6 


7. 


8 


5 


8. 


17 





16. 


25 


5 


25. 


34 





33. 


42 


5 


42- 



50 


9 
7 
6 
4 
8 
2 
6 
1 



50 

50 



4033 
50l41 

19 



40 12 
50!l5 



6 
7 
8 
9 

10 
20 
30 
40 
50 



8 
6 
5 
3 
6 

3 
6 

18 
1.8 
2.1 
2.4 
2.8 
3.1 
61 
9.2 
12.3 
15-4 



18 
18 

2.1 
2.4 



P.P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
70" 71° 





1 

2 
3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
3i 
35 
36 
37 
38 
39 



40 

41 
42 
43 
41 

45 
46 
47 
48 
49_ 

50 

51 
52 
53 

54 

55 
56 
57 
58 
59 
60 



Lg. Vers, 



81821 
81839 
81857 
81875 
81893 



81911 
81929 
81947 
81965 
81983 

82001 
82019 
82037 
82055 
82073 



82091 
82109 
82127 
82145 
82163 

82181 
82199 
82217 
82235 
82252 



82270 
82288 
82306 
82324 
82342 



82360 
82378 
82396 
82413 
82431 



82449 
82467 
82485 
82503 
82520 



82538 
82556 
82574 
82592 
82609 



82627 
82645 
82663 
82681 
82698 



82716 
82734 
82752 
82769 
82787 



82805 
82823 
82840 
82858 
82876 



9-82894 
Lg. Vers 



J) 



Log. Exs, 



10.28416 
.28469 
.28521 
.28574 
•28627 



10.28680 
.28733 
.28786 
.28839 
.28892 



10.28945 
.28998 
.29051 
.29104 
•29157 



10.29210 
.29263 
.29316 
.29370 
.29423 



10.29476 
.29529 
.29583 
.29636 
•29689 



10-29743 
.29796 
.29850 
.29903 
.29957 



10-30010 
.30064 
.30117 
.30171 
.30225 



10.30278 
.30332 
.30386 
.30440 
.30493 



10-30547 
.30601 
.30655 
.30709 
.30763 



10-30817 
-30871 
•30925 
.30979 
-31033 



10-31087 
•31141 
.31195 
.31249 
•31303 



10^31358 
.31412 
.31466 
.31521 
-31575 



2> 



10.31629 



J> I Log. Exs. 



53 
52 
53 
53 
52 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 
53 
53 
53 
53 
53 

53 
53 
53 
54 
53 

53 
54 
53 
54 
53 

54 
53 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 



Lg. Vers. 



9-82894 
.82911 
.829'?9 
.82947 
-82964 



9-82982 

.83000 

.83017 

83035 

.83053 



9.83070 
.83088 
.83106 
.83123 
.83141 



9-83159 
.83176 
.83194 
.83211 
-83229 



9.83247 
.83264 
.83282 
.83299 
.83317 



9.83335 
.83352 
.83370 
.83387 
•83405 



9 •83422 
.83440 
.83458 
.83475 
.8-3493 



9.83510 
.83528 
.83545 
.83563 
-83580 



9-83598 
•83615 
.83633 
.83650 
.83668 



9-83685 
.83703 
.83720 
.83737 
-83755 



9-83772 
-83790 
.83807 
-83825 
-83842 



J) 



9.83859 
.83877 
.83894 
83912 
83929 



9-83946 



Lg. Vers, 



17 
17 
18 
17 

18 
17 
17 
18 
17 

17 
18 
17 
17 
17 

18 

17 
IZ 
17 
18 

17 
17 
17 
17 
18 

17 
17 
17 
17 
17 

17 
17 
18 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
IZ 
17 

17 

17 

17 
17 
17 

17 



I) 



Log. Exs, 



10-31629 
.31684 
.31738 
.31793 
.31847 



10-31902 
.31956 
.32011 
.32066 
.32120 



10.32175 
.32230 
.32284 
.32339 
•32394 



10^32449 
•32504 
•32558 
.32613 
-32668 



10-32723 
.32778 
.32833 
.32888 
•32944 



10^32999 
.33054 
.33109 
.33164 
.33220 



10.33275 
.33330 
.33385 
.33441 
.33496 



10.33552 
.33607 
.33663 
.33718 
.33774 



10-33829 
.33885 
.33941 
•33996 
•34052 



10 •34108 
.34164 
.34220 
•34275 
•34331 



10-34387 
•34443 
•34499 
•34555 
-34611 



10.34667 
.34723 
.34780 
•34836 
•34892 



10-34948 



Log. Exs. 



2> 

54 
54 
54 
54 

54 
54 
54 
55 
54 
54 
55 
54 
55 
54 

55 
55 
54 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 

55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
56 
55 
55 
56 

55 
56 
56 
55 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 





1 
2 
3 

_4 

5 
6 
7 
8 
_9 

10 

11 

12 

13 

JA 

15 
16 
17 
18 
19 

20 

21 
22 
23 
_24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 

89. 
40 

41 
42 
43 
44 

45 
46 
47 
48 



50 

51 
52 
53 
54 

55 
56 
57 
58 

60 



P.P. 



56 



6 

7 
8 
9 
10 
20 
30 
40 
50 



5 


6 


6 


6 


7 


5 


8 


5 


9 


4 


18 


g 


28 


2 


37 


6 


47 


1 



6 


5.5 


7 


6.5 


8 


7.4 


9 


8.3 


10 


9.2 


20 


18.5 


30 


27^7 


40 


37-0 


50 


46-2 



54 



6 

7 
8 
9 
10 
20 
30 
40 
50 



5.4 


[ 5- 


6.3 


6. 


7.2 


7- 


8-2 


8. 


9-1 


9- 


18-1 


18- 


27-2 


27. 


36-3 


36- 


45-4 


45. 





53 


6 


5.3 


7 


6.2 


8 


7.1 


9 


8.0 


10 


8.9 


20 


17.8 


30 


26-7 


40 


35-6 


50 


44.6 



56 

5-6 

6.5 

7-4 

8-4 

9.3 

18.6 

28.0 

37.3 

46.6 

55 

5-5 

6-4 

7-3 

8.2 

9.1 

18-3 

27-5 

36-6 

45-8 

54 

4 
3 
2 
1 






53 

5-3 

6-2 

7-0 

7-9 

8-5 

17.6 

26-5 

35-3 

44-1 



53 

5.2 

6-1 

7-0 

7-9 

8.7 

17-5 

26-2 

35-0 

43-7 





18 


1^ 


6 


1.8 


1-7 


7 


2-1 


2-0 


8 


2-4 


2-3 


9 


2.7 


2-6 


10 


3-0 


2-9 


20 


6-0 


5-8 


30 


9.0 


8^7 


40 


12-0 


11-6 


50 


15.0 


14.6 



17 

1^7 
2^0 
2-2 
2-5 
2-8 
5-6 
8-5 
11-3 
14-1 



P. P. 



627 



TABLE VIII.— LOGARITHMIC VERSED SINES AND "EXTERNAL SECANTS, 



73' 



73' 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 



30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

81 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



Lg. Vers. 



83946 
83964 
83981 
83999 
84016 



84033 
84051 
84068 
84035 
84103 



84120 
84137 
84155 
84172 
84189 

84207 
84224 
84241 
84259 
84276 



84293 
84310 
84328 
84340 
84362 



84380 
84397 
84414 
84431 
84449 



84466 
84483 
84500 
84517 
84535 



84552 
84569 
84586 
84603 
84620 



84638 
84655 
84672 
84689 
84706 



84724 
84741 
84758 
84775 
84792 



84809 
84826 
84844 
84861 
84878 



84895 
84912 
84929 
84946 
84963 



84980 



Lg. Vers. 



J> 



17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 



£> 



Log.Exs. 



10.34948 
.35005 
-35061 
.35117 
.35174 



10.35230 
.35286 
.35343 
.35399 
.35456 



10.35513 
.35569 
.35626 
.35683 
.35739 



10.35796 
.35853 
.35910 
.35967 
.36023 



10.36080 
.36137 
.36194 
.36251 
.36308 



10 



.36366 
.36423 
.36480 
.36537 
.36594 



10.36652 
.36709 
.36766 
.36824 
.36881 



10-36938 
.36996 
.37054 
.37111 
•37169 



10.37226 
.37284 
.37342 
.37399 
.37457 



10.37515 
-37573 
-37631 
-37689 
•37747 



10-37805 
-37863 
.37921 
.37979 
-38037 



10.38095 
-38153 
-3821? 
.38270 
■38328 



10-3«387 
Log.Exs. 



D 

56 
56 
56 
56 

56 
56 
56 
56 
57 

56 
56 
56 
57 
56 

57 
56 
57 
57 
56 

57 
57 
57 
57 
57 

57 
57 
57 
57 
57 

57 
57 
57 
57 
57 

57 
57 
58 

Vi 

57 
57 
58 
57 
58 

58 
57 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 



Lg. Vers 



84980 
84997 
85014 
85031 
85049 



85066 
85083 
85100 
85117 
85134 



85151 
85168 
85185 
85202 
85219 



85236 
85253 
85270 
85287 
85304 



85321 
85338 
85355 
85372 
85389 



85405 
85422 
85439 
85456 
85473 
85490 
85507 
85524 
85541 
85558 



85575 
85592 
85608 
85625 
85642 



85659 
85676 
85693 
85710 
85726 



85743 
85760 
85777 
85794 
85811 



85827 
85844 
85861 
85878 
85895 



85911 
85928 
85945 
85962 
85979 



o «f^P95 



^^iLg.Vers. 



17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 
17 
17 
17 
17 
17 

16 
17 
17 
17 
17 

17 
17 
16 
17 
17 

17 

17 
16 
17 
17 

17 
ll 
17 
17 
16 

17 
17 
16 
17 
17 

16 
17 
17 
16 
17 
16 

w 

16 
17 
16 



2> 



Log. Exs. 



10.38387 
•38445 
.38504 
.38562 
-38621 



10-38679 
.38738 
.38796 
.38855 
.38914 



10.38973 
.39031 
.39090 
.39149 
.39208 



10.39267 
.39326 
.39385 
.39444 
•39503 



10-39562 
.39621 
.39681 
.39740 
-39799 



10-39859 
-39918 
.39977 
.40037 
■40096 



10^40156 
-40216 
-40275 
-40335 
-40395 



10.40454 
.40514 
.40574 
.40634 
.40694 



10.40754 
.40814 
.40874 
.40934 
.40994 



10.41054 
.41114 
.41174 
.41235 
.41295 



10-41355 
.41416 
.41476 
.41537 
.41597 



10.41658 
.41719 
.41779 
.41840 
.41901 



10.41P62 



Log. Exs. 



1> 

58 
58 
58 
58 

58 
58 
58 
59 
58 

59 
58 
59 
59 
58 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 
60 
59 
59 
60 

59 
60 
59 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
61 
60 
6(5 
61 

61 



O 

1 

2 
3 
4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
JLi 
15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 

-§i 
40 

41 
42 
43 
44 



45 
46 
47 
48 
49 

50 

51 
52 
53 
54 



55 
56 
57 
58 
59 
60 



P.P. 



6 


6-1 


7 


7-1 


8 


8. 


9 


9. 


10 


10. X 


20 


20-3 


30 


30-5 


40 


40.6 


50 


50-8 





60 


6 


6-0 


7 


7-0 


8 


8-0 


9 


9-0 


10 


10.0 


20 


20.0 


30 


30.0 


40 


40.0 


50 


50.0 





59 


6 


5.9 


7 


6.9 


8 


7-8 


9 


8-8 


10 


9-8 


20 


19-6 


30 


29-5 


40 


39.3 


50 


49-1 



6 


5-8 


7 


6-7 


8 


7-7 


9 


8-7 


10 


9-6 


20 


19.3 


30 


29^0 


40 


38-6 


50 


48.3 





57 


6 


5-7 


7 


6-6 


9 


7.6 


9 


8.6 


10 


9.5 


20 


19-0 


30 


28.5 


40 


38-0 


50 


47.5 



60 

6-0 

7-0 

8.0 

91 

10.1 

20-1 

30-2 

40.3 

50.4 

59. 

5-9 

6-9 

7.9 

8-9 

9.9 

19-8 

29.7 

39-6 

49.6 

58^ 

5.8 

6.8 

78 

8-8 

9-7 

19-5 

29-2 

39-0 

48.7 

57 

5.7 

6-7 

7-6 

8-6 

9-6 

19-1 

28-7 

38.3 

47.9 

56 

5.6 

6.6 

7-5 

8.5 

9-4 

18-8 

28-2 

37-6 

47.1 





17 


17 


6 


1-7 


1-7 


7 


2 





2-0 


8 


2 


3 


2-2 





2 


6 


2-5 


10 


2 


9 


2-$ 


20 


5 


8 


5-6 


30 


8 


7 


8-5 


40 


11 


6 


11.3 


50 


]4 


B 


]4.1 



16 

1-g 
1-9 
2-2 
2-5 
2-7 
5-5 
8-3 
11.0 
13. 7 



P. P. 



028 



5'ABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



74= 



75^ 



O 

1 
2 
3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 



Lg. Vers, 



9.85995 
•86012 
.86029 
.86046 
.86062 



9-86079 
.86096 
.86113 
.86129 
.86146 



9.86163 
.86179 
.86196 
.86213 
.86230 



9.86246 
.86263 
.86280 
.86296 
.86313 



9-86330 
.86346 
.86363 
.86380 
.86396 



9.86413 
•86430 
.86446 
.86463 
.86479 



9.86496 
.86513 
.86529 
.86546 
.86562 



9.86579 
.86596 
.86612 
.86629 
.86645 



40 

41 
42 
43 
44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 
56 
57 
58 
59 
60 



9.86662 
.86678 
.86695 
.86712 
.86728 



9.86745 
86761 
86778 
86794 
86811 



9.86827 

.86844 

.86860 

.86877 

86893 



9-86910 
.86926 
.86943 
.86959 
-86976 



9-86992 
Lg. Vers. 



D 

17 
16 
IZ 
16 

17 

16 
IZ 
16 
17 

1§ 
16 
17 
16 
17 

16 
16 
17 
11 
16 

17 

IZ 
16 

16 
17 
16 
16 
16 

17 
16 
16 
16 
16 

IZ 

16 
16 
1§ 
16 

16 
16 

IZ 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
3 



Log. Exs. 



10-41962 
•42022 
•42083 
•42144 
.42205 



10.42266 
.42327 
•42388 
•42450 
.42511 



10.42572 
.42633 
.42895 
.42756 
•42817 



10.42879 
.42940 
.43002 
•43083 
•43125 



10.43187 
.43249 
•43310 
.43372 
•43434 



10.43496 
.43558 
•43620 
•43682 
•43744 



10-43806 
•43868 
•43931 
•43993 
•44055 



10.44118 
•44180 
•44242 
•44305 
•44368 



10-44430 

.44493 
.44556 
.44618 
•44681 



10 



•44744 
.44807 
.44870 
.44933 
•44996 



10-45059 
.45122 
.45185 
.45248 
.45312 



10.45375 
.45439 
.45502 
.45565 
.45629 



10-45693 
Log. Exs. 



2> 

60 
61 
61 
61 
61 
61 
61 
61 
61 

61 
61 
61 
61 
61 

6l 
6l 
6l 
61 
62 

6l 
62 
6l 
62 
6l 

62 
62 
62 
62 
62 
62 
62 
62 
62 
62 
62 
62 
62 
62 
63 

62 
62 
63 
62 
63 
62 
63 
63 
63 
63 

63 
63 
63 
63 
63 

63 
63 
63 
63 
63 

64 



Lg. Vers. 



9.86992 
.87009 
.87025 
.87042 
.87058 



9.87074 
.87091 
•87107 
•87124 
.87140 



9.87157 
•87173 
.87189 
•87206 
•87222 



9 •87239 
•87255 
.87271 
.87288 
.87304 



9-87320 
•87337 
•87353 
•87370 
•87386 



9.87402 
.87419 
.87435 
.87451 
.87468 



9.87484 
.87500 
.87516 
.87533 
.87549 



9.87565 
.87582 
.87598 
.87614 
.87631 



9.87647 
.87653 
.87679 
.87696 
.87712 



9.87728 
.87744 
.87761 
.87777 
-87793 



9.87809 
.87825 
.87842 
.87858 
.87874 



9.87890 
.87906 
.87923 
.87939 
.87955 



9-87971 
Lg. Vers 



Log. Exs. 



10.45693 
•45756 
•45820 
.45884 
•45947 



10.46011 
.46075 
•46139 
•46203 
.48267 



10.46331 
.46395 
•46460 
.46524 
.46588 



10.46652 
•46717 
.46781 
.46846 
•46910 



10.46975 
•47040 
.47104 
•47169 
•47234 



10-472 
.47364 
.47429 
.47494 
.47559 



10.47624 
.47689 
.47754 
.47820 
.47885 



10-47950 
.48016 
.48081 
.48147 
•48213 



10.48278 
.48344 
•48410 
•48476 
.48542 



10.48607 
•48674 
.48740 
•48806 
•48872 



10.48938 
.49004 
.49071 
.49137 
.49204 



10.49270 
.49337 
.49403 
.49470 
.49537 



10-49604 



Log. Exs, 



o 

1 

2 
3 
4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 

22 
23 
24 

25 
26 
27 
28 
_29 
30 
31 
32 
33 
-34 

35 
36 
37 
38 
39 

40 

41 
42 
43 

M. 
45 
46 
47 
48 

j49 

50 

51 
52 
53 
_54 
55 
56 
57 
58 
19 
60 



P. P. 





67 


66 


6 


6.7 


6.6 


7 


7.8 


7.7 


8 


8^9 


8.8 


9 


10-0 


10^0 


10 


11-1 


11^1 


20 


22-3 


22^1 


30 


335 


33.2 


40 


44-6 


44.3 


50 


55.8 


55.4 



66 

66 
7.7 
8.8 

9.9 
11-0 
22.0 
330 
44.0 
55.0 



65 

6.5 



65 

6.5 

76 

8-6 

9-7 
10-8 
21-6 
32.5132 
43.3143 
54:ii53 



64 

6^4 
7-5 
8^6 
9.7 





64 


63 


6J 


6 


6-4 


6.3 


6- 


7 


7 


4 


7 


4 


7- 


8 


8 


5 


8 


4 


8- 


9 


9 


g 


9 


5 


9- 


10 


10 


g 


10 




10- 


20 


21 


3 


21 


1 


21- 


30 


32 





31 


7 


31. 


40 


42 


6 


42 


3 


42 


50 


53 


3 


52 


9 


52. 





62 


62 


6 


6-2 


6-21 


7 


7.3 


7 


2 


8 


8.3 


8 


2 


9 


9.4 


9 


3 


10 


10.4 


10 


3 


20 


20.8 


20 


G 


30 


31-2 


31 





40 


41.6 


41 


3 


50 


52.1 


51 


6 



61 

6.1 

7-2 

8.2 

9-2 

10.2 

20.5 

30.7 

41.0 

51.2 



6 

7 

8 

9 

10 

20 

30 

40 

50 



61 

6-1 

7.1 

8-1 

9.1 

10. 1 

20.3 

30.5 

40.6 

50.8 



60 

6.0 

7.0 

8.0 

9.1 

10.1 

20.1 

30^i 

40-3 

50.4 



17 


1^. 


1-7 


1-61 


2.0 


1 


9 


2.2 


2 


2 


2-5 


2 


5 


2-8 


2 


7 


5-6 


5 


5 


8.5 


8 


2 


11.3 


11 





14.1 


13 


7' 



16 

1-6 
1-8 
2-1 
2-4 
2-6 
5-3 
8.0 
10.6 
13-3 



P. P. 



629 



TABLE VIII.—LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
76° 77° 






9.87971 


1 


.87987 


2 


.88003 


3 


.88020 


4 


.88036 


5 


9.88052 


6 


.88068 


7 


.88084 


8 


.88100 


9 


.88116 



10 

11 

12 
13 
14 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 



80 
31 

32 
33 
34 

35 
36 
37 
38 
39 



Lg. Vers, 



9.88133 
.88149 
.88165 
.88181 
.88197 



9.88213 
.88229 
.88245 
.88261 
.88277 



9.88294 
88310 
88326 
88342 
88358 



9.88374 

88390 

•88406 

.88422 

•88438 



9.88454 
.88470 
.88486 
.88502 
.88518 



9.88534 
.88550 
.88560 
.88582 
.88598 



40 

41 
42 
43 
44 
45 
46 
47 
48 
49^ 

50 

51 
52 
53 
54 

55 
56 
57 
5» 
5i 
60 



9-88614 
.88630 
.88646 
.88662 
.88678 



9.88694 

.88710 

88726 

88742 

88758 



9.88774 
.88790 
.88805 
.88821 
.88837 



9-88853 

.88869 

-88885 

-8890] 

88917 



9 88933 



D 

16 
16 
16 
16 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
15 
16 

16 
16 
16 
16 
16 

16 
16 
15 
16 
16 

16 
16 
15 
16 
16 

16 



Log.Exs. 



10-49604 
.49670 
•49737 
.49804 
.49871 



10.49939 
.50006 
•50073 
•50140 
.50208 



10.50275 
•50342 
.50410 
•50477 
.50545 



10.50613 
.50681 
•50748 
•50816 
.50884 



10.50952 
.51020 
.51088 
•51157 
.51225 



10.51293 
•51361 
•51430 
.51498 
.51567 



10.51636 
•51704 
.51773 
•51842 
.51911 



10.51980 
•52049 
•52118 
•52187 
.52256 



10.52325 
•52394 
•52464 
.52533 
-52603 



10.52672 
.52742 
•52812 
•52881 
.52951 



Lg.Vers.|l> 



10.53021 
•53091 
•53161 
•53231 
.53301 



10.53372 
. 53442 
.53512 
.53583 
.53653 



10 53724 



Log.Exs. 



D 

66 
67 
67 
67 

67 
67 
67 
67 
67 
67 
67 
67 
67 
68 

67 
68 
67 
68 
68 

68 
68 
68 
68 
68 

68 
68 
68 
68 
68 

69 
68 



69 

69 
69 
69 
69 
69 

69 
69 
69 
69 
69 

69 
70 
69 
69 
70 

70 
70 
70 
70 
70 

70 
70 
70 
70 
70 

70 



J> 



Lg. Vers, 



88933 
88949 
88964 
88980 
88996 



89012 
89028 
89044 
89060 
89075 



89091 
89107 
89123 
89139 
89155 



89170 
89186 
89202 
89218 
89234 



89249 
89265 
89281 
89297 
89312 



89328 
89344 
89360 
89376 
89391 



9. 



89407 
89423 
89438 
89454 
89470 



9. 



89486 
89501 
89517 
89533 
89548 



9.89564 
.89580 
.89596 
.89611 
.89627 



9.89643 
.89658 
.89674 
•89690 
.89705 



9.89721 
•89737 
•89752 
.89768 
.89783 



9-89799 
.89815 
.89830 
.89846 
-89882 



9-89877 



jLg. Vers 



16 
15 
16 
16 

16 
15 
16 
16 
15 

16 
16 
15 
16 
16 
15 
16 
15 
16 
16 
15 
16 
15 
16 
15 

16 
15 
16 
16 
15 

15 
16 
15 
16 
15 

16 
15 
16 
15 
15 

16 
15 
16 
15 
15 

16 
15 
15 
16 
15 

15 
16 
15 
15 
15 

16 
15 
15 
15 
16 
15 



J) 



Log.Exs. 



10-53724 
•53794 
.53865 
•53936 
.54007 



10.54078 
•54149 
.54220 
•54291 
.54362 



10.54433 
.54505 
.54576 
• 54647 
.54719 



10.54791 
•54862 
•54934 
•55006 
.55078 



10.55150 
•55222 
•55294 
.55366 
.55438 



10.55511 
•55583 
•55655 
•55728 
.55801 



10.55873 
.55946 
•56019 
•56092 
.56165 



10.56238 
•56311 
•56384 
•56457 
-56531 



10-56604 
•56678 
•56751 
.56825 
-56899 



10-56973 
.'57047 
•57120 
.57195 
.57269 



10.57343 
.57417 
•57491 
•57566 
.57640 



10.57715 
.57790 
•57864 
•57939 
•58014 



10.58089 



Log.Exs, 



D 

70 
71 
70 
71 

71 
71 
71 
71 
71 

71 
71 
71 
71 
72 

71 
7l 
72 
71 
72 

72 
72 
72 
72 
72 

72 
72 
72 
73 
72 

72 
73 
72 
73 
73 

73 
73 
73 
73 
73 
73 
73 
73 
74 
73 

74 
74 
73 
74 
74 

74 
74 
74 
74 
74 

75 
74 
74 
75 
75 
75 



I) 





1 
2 
3 
4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

_14 

15 
16 
17 
18 
19 

30 

21 

22 

23 

-24 

25 
26 
27 
28 
-29 

30 

31 
32 
33 
^34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
j49 

50 

51 

52 

53 

_34 

55 
56 
57 
58 
_59 

60 



P.P. 



6 

7 
8 
9 
10 
20 
30 
40 
50 



75 74 



7.5 


7-4 


8.7 


8-6 


10.0 


9.8 


11.2 


11.1 


12.5 


12.3 


25.0 


24.6 


37.5 


37.0 


50.0 


49.3 


62.5 


61.6 



73 

7.3 
85 
9-7 
10-9 
12.1 
24-3 
36.5 
48.6 
60.8 





72 


71 


70 


6 


7.2 


7.1 


7.0 


7 


8.4 


8.3 


8 


2 


8 


9.6 


9-4 


9 


4 


9 


10.8 


10-6 


10 




10 


12-0 


11.8 


11 


7 


20 


24-0 


23.6 


23 


3 


30 


36.0 


35.5 


35 


2 


40 


48.0 


47-3 


47 





50 


60.0 


59.1 


58 


7 



69 

9 

2 
3 
5 

5 

5 



6 


6. 


7 


8. 


8 


9. 


9 


10- 


10 11. 


20 23. 


30 34. 


40 46. 


50 57. 


6 


7 


8 


9 


10 


20 


30 


40 




50 



68 

6-8 



67 

6.7 
7.8 
8.9 
10.0 
11.1 
22.3 
33.5 
44.6 
55-5 



66 

6.6 

7-7 

8.8 

9-9 

11.0 

22.0 

33.0 

44.0 

55.0 



O 

0.0 





16 


16 


6 


1-6 


1.61 


7 


1.9 


1 


8 


8 


2-2 


2 


1 


9 


2-5 


2 


4 


10 


2-7 


2 




20 


5.5 


5 


? 


30 


8.2 


8 





40 


11.0 


10 


6 


50 


13.7 


13 


3 



Q 
0.0 
0.1 
0.1 
O.I 
0.2 
0.3 
0.4 



15 

1-5 
1-8 
2.0 
2.3 
2-6 
5-1 
7.7 
10.3 
12.9 



P. P. 



630 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

78° 79° 



Lg. Vers. 



55 9 

56 

57 

58 

59 



89877 
89893 
89908 
89924 
89939 



89955 
89971 
89986 
90002 
90017 



90033 
90048 
90064 
90080 
90095 

90111 
90126 
90142 
90157 
90173 



90188 
90204 
90219 
90235 
90250 

90266 
90281 
90297 
90312 
90323 



90343 
90359 
90374 
90389 
90405 

90420 
90436 
90451 
90467 
90482 



90497 
90513 
90528 
90544 
90559 



90574 
90590 
90605 
90621 
90635 



90651 
90667 
90682 
90697 
90713 



90728 
90744 
90759 
90774 
90790 



9-90805 
Lg. Vers. 



Log.Exs. 


10 


.58089 
.58164 
.58239 
.58315 
.58390 


10 


•58465 
.58541 
.58616 
.58692 
•58768 


10 


53844 
58920 
58995 
59072 
59148 


10 


59224 
59300 
59377 
59453 
59530 


10 


59306 
5S883 
59760 
59337 

59914 


10. 


59391 
60338 
60145 
60223 
60300 


10 


60378 
60455 
605331 
60611 
60638 


10. 


60766 
60844 
60923 
61001 
61079 


10. 


61158 
61236 
61315 
61393 
61472 


10 


61551 
81630 
61709 
61788 
61867 


10 


61947 

62026 

.62105 

.62185 

.62265 


10 


•62345 
•62424 
.62504 
.62585 
.62665 


10.627451 


•■ 


og.Exs. 



2> 

75 
75 
75 
75 

75 
75 
75 
76 
75 

76 
76 
75 
75 
76 

76 
76 
76 
76 
76 

78 
77 
76 
77 
77 

77 
77 
77 
77 
77 

77 
77 
77 
78 
77 
78 
73 
78 
78 
78 
78 
78 
78 
78 
79 

78 
79 
79 
79 
79 

79 
79 
79 
80 
79 

80 
79 
80 
80 
80 

80 



Lg. Vers. 



90805 
90820 
90835 
90851 
90866 



90881 
90897 
90912 
90927 
909^3 

90958 
90973 
90988 
91004 
91019 



91034 
91049 
91065 
91080 
91095 



91110 
91126 
91141 
91156 
91171 



91187 
91202 
91217 
91232 
91247 



91263 
91278 
91293 
91308 
91323 



91338 
91354 
91369 
91384 
91399 



91414 
91429 
91445 
91460 
91475 



91490 
91505 
91520 
91535 
91550 



91565 
91581 
91596 
91611 
91626 



91641 
91656 
91671 
91686 
91701 



91716 
Lg. Vers. 



n 



15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 

15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 
15 



Log. Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



62745 
62825 
62906 
62986 
63067 

63148 
63229 
63310 
63391 
63472 



63553 
63634 
63716 
63797 
63879 



63961 
64043 
64125 
64207 
64289 



64371 
64453 
64536 
64618 
64701 



64784 
64867 
64950 
65033 
65116 



65199 
65283 
65366 
65450 
65534 



65617 
65701 
65785 
65870 
85954 



66038 
66123 
66207 
66292 
66377 



66462 
66547 
66632 
66717 
66803 



66888 
66974 
67059 
67145 
67231 



67317 
67403 
67490 
67576 
67663 



2> 



10.67749 



Log.Exs. 



80 
80 
80 
81 
80 
81 
81 
81 
81 

81 
81 
81 
81 
81 

82 
82 
82 
82 
82 
82 
32 
82 
82 
83 
82 
83 
83 
83 
83 

83 
83 
83 
83 
84 

83 
84 
84 
84 
84 

84 
84 
84 
84 
85 

85 
85 
85 
85 
85 

85 
85 
85 
86 
86 

86 
86 
86 
86 
86 

86 



O 

1 
2 
3 
4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
2A 
25 
26 
27 
28 
21 
30 
31 
32 
33 
3i 
35 
36 
37 
38 
39 

40 

41 
42 
43 

45 
46 
47 
48 
4:9 
50 
51 
52 
53 
_5i 
55 
56 
57 
58 
59 

60 



P. P. 





86 


85 


S4 


6 


8.6 


8.5 


%. 


7 


10 





9 


9 


9. 


8 


11 


4 


11 


3 


11. 


9 


12 


9 


12 


7 


12. 


10 


14 


3 


14 




14. 


20 


28 


6 


28 


3 


28. 


30 


43 





42 


5 


42. 


40 


57 


3 


56 


6 


56. 


50 


71 


6 


70 


8 


70. 





83 


83 


8J 


6 


8.3 


8.2 


8. 


7 


9 


7 


9 


5 


9 


8 


11 





10 


9 


10. 


9 


12 


4 


12 


3 


12. 


10 


13 




13 


6 


13. 


20 


27 


6 


27 


3 


27. 


30 


41 


5 


41 





40. 


40 


55 


3 


54 


g 


54. 


50 


69 


1 


68 


3 


67. 





80 


79 


7^ 


6 


8.0 


7.9 


7. 


7 


9 


3 


9 


2 


9. 


8 


10 


6 


10 


5 


10. 


9 


12 





11 


8 


11. 


10 


13 


3 


13 


1 


13. 


20 


26 


6 


26 


3 


26. 


30 


40 


039 


5 


39. 


40 


53 


3;52 


6 


52. 


50 


66 


6,65 


8 


65. 





77 


76 


7^ 


6 


7.7 


7.6 


7 


7 


9 





8 


3 


8. 


8 


10 


2 


10 


1 


10. 


9 


11 




11 


4 


11. 


10 


12 


3 


12 


6 


12. 


20 


25 


6 


25 


3 


25. 


30 


38 


5 


38 





37. 


40 


51 


Ci 


50 


5 


50. 


50 


64 


1 


63 


3 


62. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



O 

0.0 
0.0 
0.0 
0.1 
0.1 
O.I 
0.2 
0.3 
0.4 





16 


15 , 


6 


1 6 


1 5| 


7 


1 


3 


1 


8 


8 


2 


1 


2 





9 


2 


4 


2 


3 


10 


2 


6 


2 


6 


20 


5 


3 


5 


1 


30 


8 





7 


7 


40 


10 


g 


10 


3 


50 


13 


3 


12 


9 



15 

1.5 
17 
2.0 
2.2 
2^5 
5.0 
7.5 
10.0 
12-5 



P. P. 



631 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
80° 81° 



O 

1 

2 

3 

_4 

5 

6 

7 

8 

_9 

10 

11 

12 

13 

li 

15 

16 

17 

18 

19 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
3i 
35 
36 
37 
38 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 



50 

51 
52 
53 
54 
55 
56 
57 
58 
59 

60 



Lg. Vers 



91716 
91731 
91746 
91761 
91776 



91791 
91807 
91822 
91837 
91852 



91867 
91882 
91897 
91912 
91927 



91942 
91957 
91972 
91987 
92002 



92016 
92031 
92046 
92061 
92076 



92091 
92106 
92121 
92136 
92151 



92166 
92181 
92196 
92211 
92226 



92240 
92255 
92270 
92285 
92300 



92315 
92330 
92345 
92360 
92374 



92389 
92404 
92419 
92434 
92449 



92463 
92478 
92493 
92508 
92523 



92538 
92552 
92567 
92582 
92597 



92612 



Lg. Vers. 



JD 



Log.Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



67749 
67836 
67923 
68010 
68097 



68184 
68272 
68359 
68447 
68534 



68622 
68710 
68798 
68886 
68975 



69063 
69152 
69240 
69329 
69418 



69507 
69596 
69686 
69775 
69865 



69955 
70044 
70134 
70224 
70315 



70405 
70495 
70586 
70677 
70768 



70859 
70950 
71041 
71133 
71224 



71316 
71408 
71500 
71592 
71684 



71776 
71869 
71961 
72054 
72147 



72240 
72333 
72427 
72520 
72614 



72707 
72801 
72895 
72990 
73084 



10-73178 
Log.Exs. 



D 

86 
87 
87 
87 

87 
87 
87 
87 
87 
88 
88 
88 
88 
88 

88 
88 
88 
89 
89 

89 
89 
89 
89 
89 
90 
89 
90 
90 
90 
90 
90 

91 
90 
91 

91 
91 
91 
91 
9l 

9l 
92 
92 
92 
92 

92 
92 
92 
93 
92 

93 
93 
93 
93 
93 

93 
94 
94 
94 
94 
94 

77 



Lg. Vers, 



92612 
92626 
92641 
92656 
92671 



92686 
92700 
92715 
92730 
92745 



92759 
92774 
92789 
92804 
92818 



92833 
92848 
92862 
92877 
92892 



92907 
92921 
92936 
92951 
92965 



92980 
92995 
93009 
93024 
93039 



93053 
93068 
93083 
93097 
93112 



93127 
93141 
93156 
93171 
93185 



93200 
93214 
93229 
93244 
93258 



93273 
93287 
93302 
93317 
93331 



93346 
93360 
93375 
93389 
93404 



93419 
93433 
93448 
93462 
93477 



9-93491 
Lg. Vers. 



14 
15 
.14 
15 

15 
14 
15 
14 
15 
14 
15 
14 
15 
14 

15 

li 
14 
15 
14 

15 

14 
14 
15 
14 

15 
14 
14 
15 
14 

14 
15 
14 
14 
15 

14 
14 
14 
15 
14 

14 
14 
15 
14 
14 

14 
14 
15 
14 
14 

14 
14 
14 
14 
15 

li 
14 
14 
li 
14 

14 



Log. Exs. 



10.73178 
.73273 
.73368 
.73463 
•73558 



10-73653 
.73748 
. 73844 
•73940 
-74035 



10-74131 
.74227 
. 74324 
. 74420 
.74517 



10.74613 
.74710 
.74807 
. 74905 
.75002 



10.75099 
.75197 
.75295 
•75393 
.75491 



10-75589 
.75688 
.75786 
.75885 
•75984 



10.76083 
•76182 
.76282 
.76382 
•76481 



10-76581 
•76681 
.76782 
.76882 
-76983 



10-77083 
.77184 
.77286 
.77387 
•77488 



10.77590 
.77692 
.77794 
.77896 
.77998 



10-78101 
.78203 
.78306 
.78409 
.78513 



10-78616 
.78720 
.78823 
.78927 
.79031 



10.79136 
Log. Exs. 



1> 

95 
94 
95 
95 

95 
95 
95 
96 
95 

96 
96 
96 
96 
96 

96 
97 
97 
97 
97 

97 
98 
97 
98 
98 

98 
98 
98 
99 
99 

99 
99 
99 
100 
99 

100 
100 
100 
100 
100 

100 
101 
101 
101 
101 

lOl 
102 
102 
102 
102 

102 
102 
103 
103 
103 

103 
104 
103 
104 
104 
104 



o 

1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
JL4 

15 
16 
17 
18 

ii 
20 

21 
22 
23 
24 



30 

31 
32 
33 
34 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 
45' 
46 
47 
48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 

60 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



90 

9.0 
10.5 
12.0 
13.5 
15.0 
30.0 
45.0 
60.0 
75.0 





9 


6 


0.9 


7 


1.0 


8 


1.2 


9 


1.3 


10 


1.5 


20 


3.0 


30 


4.5 


40 


6.0 


50 


7.5 





7 


6 


0.7 


7 


0.8 


8 


0.0 


9 


1.0 


10 


1.1 


20 


2.3 


30 


3.5 


40 


4.6 


50 


5.8 





5 


6 


0.51 


7 





6 


8 





6 


9 





7 


10 





3 


20 


1 


6 


30 


2 


5 


40 


3 


3 


50 


4 


1 





15 


6 


1.5 


7 


1.8 


8 


2.0 


9 


2.3 


10 


2.6 


20 


5.1 


30 


7.7 


40 


10.3 


50 


12.9 



80 
8.0 
9.3 
10.6 
12.0 
13.3 
26.6 
40.0 
53.3 
66.6 

8 
0.8 
0.9 
1.0 
1.2 
1.3 
2.6 
4.0 
5.3 
6.6 

6 

0.6 
0.7 
0.8 
0.9 
1.0 
2.0 
3.0 
4.0 
5.0 

4 

0.4 
0.4 
0.5 
0.6 
0.6 
1.3 
2.0 
2-6 
3.3 

15 

1.5 
1.7 
2.0 
2.2 
2.5 
5.0 
7.5 
10.0 
12.5 



6 

7 

8 

9 

10 

20 

30 

40 

50 



1.3 

1-7 
1.9 
2.2 
2.4 
4.§ 

9.'g 
12.1 



P. P. 



632 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



83' 



83' 



Lg.Vers, 



93191 
93506 
93520 
93535 
93549 



93564 
93578 
93593 
93607 
93622 



93636 
93651 
93665 
93680 
93694 



93709 
93723 
93738 
93752 
93767 



93781 
93796 
93810 
93825 
93839 



93853 
93868 
93882 
93897 
93911 



93925 
93940 
93954 
93969 
93983 



93997 
94012 
94026 
94041 
94055 



94069 
94084 
94098 
94112 
94127 



94141 
94155 
94170 
94184 
94198 



94213 
94227 
94241 
94256 
94270 



94284 
94299 
94313 
94327 
94341 



9-94356 



Lg. Vers 



D 



14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 
15 



n 



Log. Exs. 



10 



10 



10 



10 



iO 



10 



10 



10 



10 



10 



10 



10 



79136 
79240 
79345 
79450 
79555 



79660 
79766 
79871 
79977 
80083 



80189 
80296 
80402 
80509 
80616 



80723 
80831 
80938 
81046 
81154 



81262 
81371 
81479 
81588 
81697 



81806 
81916 
82025 
82135 
82245 

82356 
82466 
82577 
82688 
82799 



82910 
83022 
83133 
83245 
83358 



83470 
83583 
83695 
83809 
83922 



84035 
84149 
84263 
84377 
84492 



84607 
84721 
84837 
84952 
8 5068 

85183 
85299 
85416 
85532 
85649 



10.85766 



Log. Exs, 



D 



104 
105 
104 
105 

105 
105 
105 
106 
106 

106 
106 
106 
107 
107 

107 
107 
107 
108 
108 

108 

108 
108 
109 
109 

109 
109 
109 
110 
110 

110 

110 

110 

11 

11 

11 

11 

11 

112 

112 

112 
112 
112 
113 
113 

113 
114 
114 
114 
114 

115 
114 
115 
115 
116 

115 
116 
116 
116 
117 
117 



n 



Lg. Vers, 



94356 
94370 
94384 
9439b 
94413 



94427 
94441 
94456 
94470 
94484 

94498 
94512 
94527 
94541 
94555 



94569 
94584 
94598 
94612 
94626 



94640 
94655 
94669 
94683 
94697 



94711 
94726 
94740 
94754 
94768 



94782 
94796 
94810 
94825 
94839 



94853 
94867 
94881 
94895 
94909 



94923 
94938 
94952 
94966 
94980 



94994 
95008 
95022 
95036 
95050 



95064 
95078 
95093 
95107 
95121 



95135 
95149 
95163 
95177 
95191 



9 95205 
Lg. Vers. 



D 

14 

14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 

"d 



Log. Exs. 



10.85766 
85884 
86001 
86119 
86237 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



86355 
86474 
86592 
86711 
86831 



86950 
87070 
87190 
87310 
87431 



87552 
87673 
87794 
87916 
88038 



88160 
88282 
88405 
88528 
88651 



88775 
88898 
89022 
89147 
89271 



89396 
89521 
89647 
89773 
89899 



90025 
90152 
90279 
90406 
90533 



90661 
90789 
90917 
91046 
91175 



91304 
91434 
91564 
91694 
91825 



91956 
92087 
92218 
92350 
92482 



92614 
92747 
92880 
93014 
93147 



93281 



Log. Exs. 



117 
117 
117 
118 

118 
118 
118 
119 
119 

119 
120 
120 
120 
120 

121 
121 
121 
121 
122 

122 
122 
122 
123 
123 

124 
123 
124 
124 
124 

125 
125 
125 
126 
126 

126 
126 
127 
127 
127 

128 
127 
128 
129 
129 

129 
130 

129 
130 
130 

131 
131 
131 
131 
132 

132 
133 
133 
133 
133 

134 





1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
_14 

15 
16 
17 
18 
19 

20 

21 
22 
23 

-24 
25 
26 
27 
28 

-29 

30 
31 
32 
33 

-34 

35 
36 
37 
38 
-39 
40 
41 
42 
43 

45 
46 
47 
48 
j49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
5SL 
60 



P.P. 





130 


6 


13.01 


7 


15 


\ 


8 


17 


3 


9 


19 


5 


10 


21 


g 


20 


43 


3 


30 


65 





40 


86 


6 


50 


108 


3 



110 



6 11.0 


7 12.8 


8 14.6 


9 


16-5 


10 


18.3 


20 


36.6 


30 


55.0 


40 


73-3 


50 


91.6 



3 

0.3 



1.0 
1.5 
2.0 
2.5 





1 


6 


0.11 


7 





1 


8 





1 


9 







10 







20 





3 


30 





5 


40 





6 


50 





8 



130 

12.0 
14.0 
16.0 
18.0 
20.0 
40.0 
60.0 
80.0 
100.0 



100 

10.0 
11.6 
13.3 
15-0 
16-6 
33-3 
50.0 
66.6 
83.3 



3 

0.2 
0.2 
0.2 
0.3 
0.3 
0.6 
l.Q 
1.3 
1-6 



O 

o.s 

0.0 



6 

7 
8 
9 
10 
20 
30 
40 
50 



14 

1.4 
1.7 
1.9 
2.2 
2.4 
4.8 
7.2 
9.6 
12.1 



0.4 



14 

1.4 
1.6 
1.8 
2.1 
2-3 
4.6 
7.0 
9.3 
11.6 



P.P. 



633 



TABLE VIII.—LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
84* 85° 



Lg. Vers. 



9.95205 
.95219 
.95233 
•95247 
.95261 



9-95275 
.95289 
•95303 
.95317 
•95331 



9.95345 
.95359 
.95373 
•95387 
•95401 



9^95415 
•95429 
•95443 
■95457 
•95471 



95485 
95499 
95513 
95527 
95540 



O 

1 
2 
3 

_4 

5 

6 

7 

8 
^ 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
M 
45 
46 
47 
48 
49 

50 9^95901 



95554 
95568 
95582 
95596 
95610 



95624 
95638 
95652 
95666 
95680 



I> 



9. 95693 
95707 
95721 

9573!i 
95749 



9^95763 
95777 
95791 
95804 
95818 



9^95832 
.95846 
•95860 
.95874 
•95888 



51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



.95915 
.95929 
.95943 
.95957 



9.95970 
.95984 
.95998 
•96012 
•96026 



9 •96039 



' Lg.Vers 



14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
13 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
13 

14 
14 
14 
14 
14 

14 
13 
14 
14 
14 

13 
14 
14 
14 
14 

13 
14 
14 
13 
14 

14 
14 
13 
14 
14 

13 
14 
13 
14 
14 

13 
14 
14 
13 
14 

13 



Log.Exs. 



10.93281 
•93416 
•93551 
•93686 
.93821 



10.93957 
•94093 
.94229 
.94366 
•94503 



10 • 94641 
•94778 
.94917 
•95055 
.95194 



10.95333 
.95473 
.95613 
.95753 
•95894 



10 •96035 
•96176 
.96318 
.96461 
.96603 



10.96748 
•96889 
.97033 
.97177 
.97322 



10.97467 
.97612 
.97758 
.97904 
•98050 



1> 



10^98197 

•98345 

•98492 

• 9??640 

98789 



10.98938 
•99087 
•99237 
•99387 
•99538 



10^99689 

•99841 

10.99993 

11.00145 

•00298 



11.00451 
.00605 
•00759 
.00914 
•01069 



11^01225 
•01381 
.01537 
.01694 
•01852 



11^02010 



Log.Exs. 



134 
135 
135 
135 

135 
136 
136 
137 
137 

137 
137 
138 
138 
139 

139 
139 
140 
140 
140 

14l 
141 
142 
142 
142 

143 
143 
144 
144 
144 

145 
145 
145 
146 
146 

147 
147 
147 
148 
149 

149 
149 
150 
150 
151 

151 
151 
152 
152 
153 

153 
154 
154 
155 
155 

155 
156 
156 
157 
157 

158 



Lg.Vers. 



9 •96039 
•96053 
•96067 
•96081 
•96095 



9^96108 
96122 
96136 
96150 
96163 



9^96177 
•96191 
•96205 
•96218 
•96232 



•96246 
•96259 
.96273 
•96287 
•96301 



9 •96314 
.96328 
•96342 
•96355 
•96369 



9 •96383 

.96397 

.96410 

.96424 

96438 



9^96451 
.96465 
•96479 
•96492 
•96506 



9.96519 
.96533 
.96547 
•96560 
•96574 



96588 
•96601 
.96615 
.96629 
.96642 



9.96656 
.96669 
.96683 
.96697 
•96710 

9^96724 
96737 
96751 
96764 
96778 



D 



9^96792 
•96805 
.96819 
.96832 
.96846 



9. 96859 



14 
13 
14 
14 

13 
14 
13 
14 
13 

14 
13 
14 
13 
14 

13 
13 
14 
13 
14 

13 
14 

13 
13 
14 

13 
14 
13 
13 
14 

13 
13 
14 
1^ 
13 

13 
14 
13 
13 
14 

13 
13 
13 
14 
13 

13 
13 
13 
14 
13 

13 
13 
13 
13 
14 

13 
13 
13 
13 
13 

13 



Log.Exs. 



11 



11 



11 



11 



11 



11 



11 



11 



11 



Lg.Vers, 



11 



11 



11 



11 



02010 
02168 
02327 
02487 
02646 



02807 
02968 
03129 
03291 
03453 



03616 
03780 
03944 
04108 
04273 



04438 
04604 
04771 
04938 
05106 



05274 
05443 
05612 
05782 
05952 



06123 
06295 
06467 
06640 
06813 



06987 
07161 
07336 
07512 
07688 



07865 
08043 
08221 
08400 
08579 



08759 
08940 
09121 
09303 
09486 



09669 
09853 
10038 
10223 
10409 



10595 
10783 
10971 
11160 
11349 



11539 
11730 
11922 
12114 
12307 



12501 



158 
159 
159 
159 

160 
161 
161 
161 
162 

163 
163 
164 
164 
165 
165 
166 
167 
167 
167 

168 
169 
169 
169 
170 

171 
17l 
172 
173 
173 

174 
174 
175 
176 
176 

177 
177 
178 
179 
179 

180 
180 
18l 
182 
182 

183 
184 
185 
185 
186 

186 
187 
188 
189 

189 

190 
191 
191 
192 
193 
193 



T> Log.Exs. 



O 

1 
2 
3 
j4 

5 
6 
7 
8 

10 

11 
12 
13 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
-li 
30 
31 
32 
33 
-li 
35 
36 
37 
38 
39 

40 

41 
42 
43 

45 
46 
47 
48 
j49 

50 

51 
52 
53 

55 
56 
57 
58 
31 
60 



JJ 



P. P. 





190 


6 


19.0 


7 


22.1 


8 


25.3 


9 


28.5 


10 


31.6 


20 


63^3 


30 


95^0 


40 


126.6 


50 


158.3 



180 

180 
21-0 
24^0 
27 
30 
60 
90 
120 



170 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



17 





16- 


19 


8 


18^ 


22 


6 


21^ 


25 


5 


24 • 


28 


3 


26 • 


56 


6 


53 • 


85 





80 • 


113 


3 


106. 


141 


6 


133. 



150 

15.0 

17 

20 

22 

25 

50 

75 
100 
125 



150.0 

160 


6 
3 

6 
3 

6 
3 

140 

14^0 
16 



18 

21 
23 
46 
70 
93 
116 





130 


9 


^ 


6 


13 


0.9 


0- 


7 


15 


1 


1^0 


0. 


8 


17 


3 


1^2 


1. 


9 


19 


5 


1^3 


1. 


10 


21 


6 


1-5 


1. 


20 


43 


3 


30 


2. 


30 


65 





4^5 


4. 


40 


86 


6 


6^0 


5. 


50 


108 


3 


7^5 


6. 



6 

7 

8 

9 
10 
20 
30 3-5 
404 • 6 
50i5.8 



6 

0^6 
0^7 
0^8 
0^9 
1.0 
2.0 
30 
4.0 
5-0 



6 

7 

8 

9 

10 

20 

30 

40 

50 



14 

1.4 
1.7 
1.9 
2-2 
2.4 
4.8 
7.2 
9.6 
12.1 



14 

1-4 
1^6 
1-8 
2.] 



5 

•6 
•6 

• 7 
•8 

• 6 

• 5 

• 3 
.1 

13 



P.P. 



634 



TABLE YIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
86° 87° 



Lg. Vers. 



96859 
96837 
98887 
96900 
96914 



96927 
96941 
96954 
96968 
96981 



96995 
97008 
97022 
97035 
97049 



97062 
97076 
97089 
97103 
97116 



97130 
97143 
97157 
97170 
97183 



S7197 
97210 
97224 
97237 
97251 



97264 
97277 
97291 
97304 
97318 



97331 
97345 
97358 
97371 
97385 



97398 
97412 
97425 
97438 
97452 



97465 
97478 
97492 
97505 
97519 



97532 
97545 
97559 
97572 
97585 



97599 
97612 
97625 
97639 
97652 



9.97665 



Lg. Vers, 



D 

13 
14 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
33 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 



Log. Exs. 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



12501 
12696 
12891 
13087 
13284 



13482 
13680 
13879 
14079 
14280 



14482 
14684 
14887 
15092 
15297 



15502 
15709 
15917 
16125 
16334 



16544 
16755 
16967 
17180 
17394 



17609 
17824 
18041 
18259 
18477 



18697 
18917 
19138 
19361 
19584 



19809 
20034 
20261 
20489 
20717 



20947 
21178 
21410 
21643 
21877 



22112 
22349 
22586 
22825 
23065 



23306 
23548 
23792 
24037 
24283 



24530 
24778 
25028 
25279 
25531 
25785 



Log. Exs. 



195 
195 
196 
196 

198 
198 
199 
200 
201 

201 
202 
203 
204 
205 
205 
206 
208 
208 
209 
210 
211 
212 
213 
214 

214 
215 
216 
218 
218 

219 
220 
221 
222 
223 
224 
225 
227 
227 
228 

230 
230 
232 
233 

234 

235 

236 
237 
239 
239 

241 
242 
243 
245 
246 

247 
248 
250 
251 
252 

254 



Lg. Vers 



9.97665 
•97679 
•97692 
•97705 
•97718 



9^97732 
•97745 
•97758 
•97772 
•97785 



9 •97798 

97811 

•97825 

•97838 

•97851 



9^97864 
•97878 
•97891 
•97904 
•97917 



•97931 
•97944 
.97957 
•97970 
•97984 



9-97997 
.98010 
.98023 
.98036 
•98050 



9.98063 
.98076 
.98089 
.98102 
.98116 



.98129 
.98142 
.98155 
.98168 
•98181 



9.98195 
.98208 

.98221 
.98234 
•98247 



9 •98260 
98273 
98287 
98300 
98313 



9.98328 
•98339 
.98352 
.98365 
•98378 



9^98392 
•98405 
•98418 
•98431 
•98444 



9. 98457 



Lg. Vers. 



D 

13 
13 
13 
13 

13 
13 
13 
13 
13 
13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 



Log. Exs. 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



11 



25785 
2604C 
26297 
26554 
26814 

27074 
27336 
27599 
27864 
28]31 



28398 
28668 
28938 
29211 
29485 



29760 
30037 
30316 
30596 
30878 



31162 
31447 
31734 
32023 
32313 



32606 
32900 
33196 
33494 
33793 



2> 



34095 
34398 
34704 
35011 
35321 



35632 
35946 
36261 
36579 
36899 



37221 
37546 
37872 
38201 
38532 



38866 
39201 
39540 
39880 
40224 



40569 
40918 
41269 
41622 
41979 



42338 
42699 
43064 
43431 
43802 



44175 



Log. Exs. 



255 

256 
257 
259 

260 
262 
263 
265 
266 

267 
269 
270 
272 
274 

275 
277 
278 
280 
282 

283 
285 
287 
288 
290 

292 
294 
296 
298 
299 
301 
303 
305 
307 
309 

311 
313 
315 
318 
320 

322 
324 
326 
328 
331 

333 
335 
338 
340 
343 
345 
348 
351 
353 
356 

359 
361 
364 
367 
370 

373 



10 

11 

12 
13 
ii 
15 
16 
17 
18 
_19 
20 
21 
22 
23 
24 

25 
26 
27 
28 
2^ 

30 

31 
32 
33 
-34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
li 
50 
51 
52 
53 
li 
55 
56 
57 
58 
11 
60 



P. P. 





25 





6 


25-Oi 


7 


29 


1 


8 


33 


3 


9 


37 


5 


10 


41 


6 


20 


83 


3 


30 


125 





40 


166 


6 


50 


208 


3 



230 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 
7 
8 
9 

10 
20 
30 
40 
50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



23 





26 


8 


30 


6 


34 


5 


38 


3 


76 


6 


115 


C 


153 


3 


191 


6 



210 



21 





24 


5 


28 





31 


5 


35 





70 





105 





140 





175 






190 

19.0 
22 



25 
28 
31 
63 
95 
126 
158 



6 

7 

8 

9 

10 

20 

30 

40 

50 






2 


0^1 








0^1 





2 


0.1 





3 


0.1 





3 


0^1 





6 


0.3 


1 





0.5 


1 


3 


0.6 


1 


6 


0-8 



14 

1.4 



1 
1 
2 
2 
4 
7 
9 
50i]l 



240 

24.0 

28.0 

32.0 

36.0 

40.0 

80.0 

120.0 

160.0 

200.0 

220 

22.0 

25.6 

29.3 

33.0 

36.6 

73.3 

110.0 

146.6 

183.3 

200 

20.0 

23.3 

26-6 

30.^ 

33.3 

68.6 

100.0 

133.3 

166.6 

4 3 

0.410.3 
0.40.3 
0-5 0-4 
0-60.4 
0.60-5 
1.3 1.0 
2.01.5 
2^62.0 
3.3I2.5 



O 

0.0 
0.0 
0.0 
0-1 
0-1 
O-I 
0.2 
0^3 
0.4 

13 

1.3 
1.5 
1.7 
1.9 
2.1 
4.3 
6.5 
8-6 
ilO.8 



13 



P. P. 



635 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



88° 



89' 



Lg. Vers. 



98457 
98470 
98483 
98496 
98509 



98522 
98535 
98548 
98562 
98575 



98588 
98601 
98614 
98627 
98640 



98653 
98666 
98679 
98692 
98705 



98718 
98731 
98744 
98757 
98770 



98783 
98796 
98809 
98822 
98835 



98848 
98861 
98874 
98887 
98900 



98913 
98925 
98938 
98951 
98964 



98977 
98990 
99003 
99016 
99C29 



99042 
99055 
99068 
99081 
99093 



99106 
99119 
99132 
99145 
99158 



99171 
99184 
99197 
9920P 
99222 



99?3^ 



' Lg. Vers 



13 
13 
13 
13 

13 
13 
13 
13 
13 
13 
13 
13 
13 
13 

13 
13 
13 
13 
13 
13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
12 
13 
13 
13 

13 
13 
13 
13 

12 

13 
13 
13 
13 
12 

13 

13 
13 
13 
12 

13 
13 
13 
12 
13 
13 



Log.Exs. 



11.44175 
.44551 
.44931 
.45313 
.45699 



n 



11.46088 
.46480 
.46876 
.47275 
.47677 



n 



11.48083 
•48493 
.48906 
.49323 
.49743 



11.50168 
.50597 
.51029 
.51466 
.51906 



11.52351 
.52801 
.53255 
•53713 
•54176 



11 •54643 
.55116 
.55593 
•56076 
.56563 



11.57056 
.57554 
.58058 
.58567 
.59082 



11.59602 
.60129 
.60662 
.61202 
.61747 



11.62300 
.62859 
.63425 
•63998 
•64579 



11^65167 
.65762 
.66366 
.66978 
•67598 



11.68227 
.68865 
.69511 
.70168 
.70834 



11.71509 
.72196 
.72892 
.73600 
•74319 



11.75050 



Log.Exs. 



376 
379 
382 
386 

389 

392 
395 
399 
402 

406 
409 
413 

417 
420 

425 
428 
432 
436 
440 

445 
449 
454 
458 
463 

467 
472 
477 
482 
487 
492 
498 
504 
509 
515 

520 
527 
533 
539 
545 

552 
559 
566 
573 
581 

588 

595 
604 
611 
620 

628 
638 
646 
656 
666 

675 
686 
696 
707 
719 

73S 



J> 



Lg. Vers. 



99235 
99248 
99261 
99274 
99287 



99299 
99312 
99325 
99338 
99351 



99363 
99376 
99389 
99402 
99415 



99428 
99440 
99453 
99466 
99479 



99491 
99504 
99517 
99530 
99543 



99555 
99568 
99581 
99594 
99606 



99619 
99632 
99645 
99657 
99670 



99683 
99695 
99708 
99721 
99734 



99746 
99759 
99772 
99784 
99797 



99810 
99823 
99835 
99848 
99861 



99873 
99886 
99899 
99911 
99924 



99937 
99949 
99962 
99974 
99987 



D 



12 
13 
13 
13 
12 
13 
13 
12 
13 
12 
13 
13 
12 
13 

13 
12 
13 
12 
13 

12 
13 
13 
12 
13 
12 
13 
12 
13 
12 

13 
12 
13 
12 
13 

12 
12 
13 
12 
13 
12 
13 
12 
12 
13 
12 
13 
12 
12 
13 

12 
12 
13 
12 
12 

13 
19 
12 
12 
13 

10 00000 2f 
Lg. Vers. | I> 



Log.Exs. 



11-75050 
•75792 
•76547 
•77316 

^7_8097 

11.78892 
•79702 
.80527 
.81367 

.82223 



11.83095 
•83986 
•84894 
•85821 
.86768 



11.87735 
.88724 
•89735 
•90769 
•91829 



11-92914 
.94026 
.95167 
.96338 
.97541 



11.98777 

12.00048 

.01358 

.02707 

.04098 



12.05535 
.07020 
.08557 
.10149 
•11801 



12^13517 
.15302 
.17163 
•19106 
•21139 



12-23271 
•25511 
•27872 
•30367 
•33013 



12.35828 
.38837 
•42068 
•45557 
.49349 



12.53501 
.58089 
.63217 
.69029 
.75736 



12.83667 
.93371 

13.05877 
.23499 
•53615 



Infinity 



Log.Exs. 



2> 



742 
755 
768 
781 

795 
809 
825 
840 
856 

872 
890 
908 
927 
947 

967 

989 

1009 

1034 

1059 

1085 
1112 
1140 
1171 
1203 

1236 
1271 
1309 
1349 
1391 

1436 
1485 
1537 
1592 
1652 

1716 
1785 
1861 
1943 
2033 

2131 
2240 
2361 
2495 
2645 
2815 
3009 
3231 
3489 
3791 

4152 
4588 
5127 
5812 
6707 

7931 

9704 

12506 

17621 

80116 





1 

2 

3 

_4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
_li 
15 
16 
17 
18 
19 

30 

21 
22 
23 
-24 
25 
26 
27 
28 
2^ 

30 

31 
32 
33 
34 



35 
36 
37 
38 
39 

40 

41 
42 
43 
j44 

45 
46 
47 
48 
49 



50 

51 
52 
53 
54 



2> 



60 



P.P. 



I 




P.P. 



C3G 



TABLE IX.~NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 

o°-io° 



00174 



00349 



O 0533 



Sin. 



0000 29 

29 
29 
29 
29 
29 



0.0029 
0058 
0.0087 
00116 
0.0145 



0.0203 
0.0232 
0.0262 
0.0291 
00320 



0.0378 
00407 
00436 
00465 
0-0494 



0552 
0581 
0610 
0639 
0668 



0697 



0-0726 
00755 
0.0784 
00813 
00842 



o 

10 
20 
30 
40 
50 

1 
10 
20 
30 
40 
50 

2 O 
10 
20 
30 
40 
50 

3 
10 
20 
30 
40 
50 

4 
10 
20 
30 
40 
50 

5 

10 
20 
30 
40 
50 

6 
10 
20 
30 
40 
50 

7 
10 
20 
30 
40 
50 

8 O 
10 
20 
30 
40 
50 

9 
10 
20 
30 
40 
50 

10 Oio 1736 



Tan. 



0000 



00174 



0871 



0.0900 
0.0929 
0.0958 
0.0987 
0.1016 



O 1045 



1074 
1103 
1132 
1161 



0.1190 



0. 1318 



1391 



1247 
1276 
1305 
1334 
1363 



0.1420 
0.1449 
0.1478 
0.1507 
0.1535 



O 1564 



0.1593 
0.1622 
0.1650 
0.1679 
0.1708 



29 

29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 

29 

29 

29 

29 

29 

29 

28 

29 

29 

29 

29 

28 

29 

29 

29 

28 

29 

28 

29 

29 

28 

29 

28 

29 

28 

29 

28 

28 

29 

28 



00349 

0-0378 
0.0407 
0-0436 
0-0486 
0495 



00534 



0-0029 
0.0058 
0.0087 
0.0116 
0.0145 



0.0203 
0.0233 
0.0262 
0.0291 
0.0320 



0-0553 
0-0582 
00611 
0-0641 
00670 



O 0699 



0-0728 
00758 
0-0787 
0-0816 
0845 



00875 



0.0904 
0.0933 
0.0963 
0.0992 
0-1021 



Cot. 



01051 



0.1080 
0.1110 
0.1139 
0.1169 
0-1198 



1338 



0-1257 
0.1287 
0.1316 
0.1346 
0.1376 



01405 



0.1435 
0.1465 
0.1494 
0-1524 
0.1554 



O 1584 



Cos, 



d. 



0.1613 
0.1643 
0.1673 
0-1703 
0.1733 



1763 



29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

29 

2 

2 

2 

29 
29 
29 
29 
29 
29 
29 
29 
29 
30 
29 
29 
30 
29 
30 
29 
30 
29 
30 
30 
30 
30 
30 



QC 



343-773 
171-885 
114-588 
85-9398 
68 7501 



57 3899 



49-1039 
42.9641 
38.1884 
34-3677 
31-2416 



38 6363 



26-4316 
24.5417 
22.9037 
21.4704 
20. 2055 



190811 



18.0750 
17.1693 
16.3493 
15.6048 
14.9244 



143006 



13.7267 
13.1969 
12.7052 
12-2505 
11-8261 



11. 4300 



11-0594 
10-7119 
10-3854 
10-0780 
9-7881 



9 5143 



Cot. 



9-2553 
9.0098 
8.7769 
8.5555 
8-3449 



8 1443 



7.9530 
7.7703 
7-5957 
7- 4287 
7.2687 



7 1153 



6.9682 
6.8269 
6.6911 
6.5605 
6-4348 



6 3137 



d. 



6-1970 
6-0844 
5-9757 
5-8708 
5-7693 
5. 6713 



Tan. 



1860 
1398 
7756 
8207 
1261 
6053 
2046 
8898 
6380 
4333 
2648 
1244 
0061 
9056 
8195 
7450 
6804 
6237 
5739 
5298 
4907 
4557 
4243 
3961 

3706 

3475 
3265 
3073 
2899 
2738 
2590 
2454 
2329 
2213 
2106 
2006 
1913 
1826 
1746 
1870 
1599 
1534 
1471 
1413 
1358 
1306 
1257 
1211 
1167 
1126 
1087 
1049 
1014 
980 



Cos. 



10000 



.0000 
■ 0000 
.9999 
-9999 
9999 



09998 



0-9998 
0-9997 
0.9996 
0.9996 
0-9995 



9994 

0-9993 



9991 
9990 
9989 
9988 



09986 

0-9984 



9983 
9981 
9979 
9977 



9975 

0-9973 
0-9971 
0.9969 
0.9967 
0-9964 



0-9962 



0-9959 
0-9956 
0-9954 
0-9951 
09948 



9945 

0-9942 
0-9939 
0-9935 
0-9932 
0-9929 



9935 

0-9922 
0-9918 
0-9914 
0-9910 
0-9906 



9902 



0-9898 
0-9894 
0-9890 
0-9886 
0-9881 



0-9877 

0-9872 
0.9867 
0.9863 
09858 
0-9853 



0-9848 



Sin. 



88 



87 



86 



O 90 

50 
40 
30 
20 
10 

89 
50 
40 
30 
20 
10 


50 
40 
30 
20 
10 


50 
40 
30 
20 
10 


50 
40 
30 
20 
10 

85 

50 
40 
30 
20 
10 

O 84 
50 
40 
30 
20 
10 

83 
50 
40 
30 
20 
10 

O 82 
50 
40 
30 
20 
10 

O 81 
50 
40 
30 
20 
10 
O SO 



P. P. 



30 39 



3 





2 


9 


2- 


6 





5 


9 


5- 


9 





8 


8 


8- 


12 





11 


8 


11- 


15 





14 


7 


14. 


18 





17 


7 


17- 


21 





20 


6 


20- 


24 





23 


6 


23- 


27 





26 


5 


26- 



29 

9 



38_ 
2 



5 4_ 

0.5J0-4 

-o'o 

-5|1 
-Oil 
-5 2 
• 02 
-5 3 
.013 
.514 





1 
1 
2 
2 
2 
^,3 
013 



3 3 



0.3 





3 





2 





0.7 





6 





5 





1.0 





9 





7 


0. 


1-4 


1 


2 


1 





0. 


1-7 


1 


5 


1 


2 




2-1 


1 


8 


1 


5 




2-4 


2 


1 


1 


7 




2-8 


2 


4 


2 







3-1 


2 


7 


2 


2 


1 . 



3 

2 
4 
6 
8 

2 
4 
6 
8 






I 





1 





3 





2 





4 





3 





g 





4 





7 





5 





9 





6 


1 








7 


1 


2 





8 


1 


3 





9 



o 

0.0 
0.1 
O.I 
0.2 
0.2 
0.3 
0.3 
0.4 
04 



P. P. 



80°-90' 



637 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 

10°-30° 



10 O 

10 
20 
30 
40 
50 

11 O 
10 
20 
30 
40 
50 

12 O 
10 
20 
30 
40 
50 

13 
10 
20 
30 
40 
50 

14 

10 
20 
30 
40 
50 

15 

10 
20 
30 
40 
50 

16 

10 
20 
30 
40 
50 

17 

10 
20 
30 
40 
50 

18 
10 
20 
30 
40 
50 

19 
10 
20 
30 
40 
50 

20 



Sin. d. Tan. J d. Cot. d. Cos. d. 



0-1736 







1908 



2079 















1765 
1793 
1822 
1851 
1879 



1936 
1965 
1993 
2022 
2050 



2107 
2136 
2164 
2193 
2221 



249 



2278 
2306 
2334 
2362 
2391 



2419 



2447 
2475 
2504 
2532 
2560 



2588 



2616 
2644 
2672 
2700 
2728 



02756 



2784 
2812 
2840 
2868 
2896 



02923 



0.2951 
0.2979 
03007 
03035 
03062 



0-3090 



0.3118 
0.3145 
0-3173 
0.3200 
0-3228 



03255 



03283 
0.3310 
03338 
03365 
03393 



03420 



28 
28 
29 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 

28 

28 

28 

28 

28 

28 

28 

28 

27 

28 

28 

27 

28 

28 

27 

28 

27 

27 

28 

27 

27 

27 

27 

27 

27 

27 
27 
27 
27 
27 



0.1763 



1944 



2125 



1793 
1823 
1853 
1883 
1913 



1974 
2004 
2034 
2065 
2095 



2156 
2186 
2217 
2247 
2278 



2308 



2339 
2370 
2401 
2431 
2462 



2493 



2524 
2555 
2586 
2617 
2648 



2679 



2710 
2742 
2773 
2804 
2836 



02867 



03057 



Cos. 



0.2899 
0-2930 
0-2962 
0-2994 
03025 



3089 
3121 
3153 
3185 
3217 



03249 



3281 
3313 
3346 
3378 
3411 



03443 



0-3476 
03508 
0-3541 
0.3574 
03607 



0-3639 

Cot. 



30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
31 
30 
31 
30 
31 
31 
30 
31 
31 
31 
3l 
31 

31 
3l 
31 
31 
31 
31 

31 

31 
31 
32 
3l 

32 

31 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
33 
33 
32 



5 1445 



5-6713 



5.5764 
5.4845 
5.3955 
5. 3093 
5.2256 



0658 
9894 
9151 
8430 
7728 



47046 



4.6382 
4.5736 
4.5107 
4 . 4494 
4.3897 



4.3315 



4-2747 
4-2193 
4-1653 
4-1125 
4-0610 



40108 



3.9616 
3.9136 
3.8667 
3 - 8208 
3-7759 



37320 



3.6891 
3 - 6470 
3-6059 
3-5655 



3-5261 



3. 4874 



-4495 
-4123 
-3759 
-3402 
-3052 



32708 



-2371 
-2040 
-1716 
-1397 
-1084 



30777 



0475 
0178 
9887 
9600 
9319 



29042 



2-8770 
2-8502 
2.8239 
2.7980 
2.7725 



27475 

Tan. 



949 
919 
890 
862 
836 
811 
787 
764 
742 
721 
701 
682 
664 

646 
629 
613 
597 
582 
568 
553 
540 
527 
515 
502 
49l 
480 
469 
458 
449 
439 

429 

420 
411 
403 
394 
387 
379 
371 
364 
357 
350 
348 
337 

331 
324 
319 
313 
307 
302 
296 
291 
286 
281 
277 
272 
267 
263 
259 
254 
250 



0.9848 



09816 



0-9781 



0-9843 
9838 
0-9832 
09827 
0-9822 



09810 
0.9805 
0.9799 
0-9793 
0.9787 



0.9775 
0-9769 
0-9763 
0-9756 
0-9750 



09743 



0-9737 
0-9730 
0-9723 
0-9717 
0-9710 



09703 



0-9696 
0-9688 
09681 
0-9674 
09666 



09659 



0-9651 
0-9644 
0-9636 
0-9628 
0-9620 



09612 



0-9604 
0.9596 
09588 
09580 
09571 



09563 



0.9554 
0.9546 
0.9537 
09528 
0-9519 



09510 



0-9501 
0-9492 
0-9483 
0-9474 
0-9464 



0-9455 



0-9445 
0-9436 
0.9426 
0-9416 
0-9407 



09397 



Sin, 



O 80 
50 
40 
30 
20 
10 

O 79 
50 
40 
30 
20 
10 

O 78 
50 
40 
30 
20 
10 

O 77 
50 
40 
30 
20 
10 

O 
50 
40 
30 
20 
10 


50 
40 
30 
20 
10 


50 
40 
30 
20 
10 

O 
50 
40 
30 
20 
10 

O 72 
50 
40 
30 
20 
10 

71 
50 
40 
30 
20 
10 

O 70 



76 



75 



74 



73 



P. P. 



33 

3-3 



32 

3.2 
4 
6 
8 

2 
4 
6 
8 



31 





30 


30 


21 


1 


3-0 


3 


2. 


2 


6 


1 


6 





5- 


3 


9 


1 


9 





8- 


7 


12 


2 


12 





11- 


5 


15 


2 


15 





14- 


6 


18 


3 


18 





17- 


7 


21 


3 


21 





20- 


8 


24 


4 


24 





23- 


9127 


4 


27 





26. 



28 28 



2 


8 


2 


8 


2. 


5 


7 


5 


6 


5- 


8 


5 


8 


4 


8- 


11 


4 


11 


2 


10- 


14 


2 


14 





13- 


17 


1 


16 


8 


16- 


19 


9 


19 


6 


18. 


22 


8 


22 


4 


21. 


25 


6 


25 


2 


24. 



10 9 



1-0 





9 


0. 


2-0 


1 


8 


1. 


3-0 


2 


7 


2. 


4-0 


3 


6 


3. 


50 


4 


5 


4- 


60 


5 


4 


4- 


7.0 


6 


3 


5. 


80 


7 


2 


6. 


9.0 


8 


1 


7. 





7 


7 


( 


B 


1 


0-7 


0-7 


0-61 


2 


1 


5 


1-4 


1 


2 


3 


2 


2 


2-1 


1 


8 


4 


3 





2-8 


2 


4 


5 


3 


7 


3-5 


3 





6 


4 


5 


4-2 


3 


6 


7 


5 


2 


4-9 


4 


2 


8 


6 





5-6 


4 


8 


9 


6 


7 


6.3 


5 


4 



27 

7 
4 
1 
8 
5 
2 
9 
6 
3 



5 

0.5 
1.0 
1.5 
2.0 
2.5 
30 
3.5 
4.0 
4.5 



P. P. 



638 



70°-80' 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS. AND COTANGENTS. 

20°-30° 



20 

10 
20 
30 
40 
50 

21 
10 
20 
30 
40 
50 

22 
10 
20 
30 
40 
50 

23 
10 
20 
30 
40 
50 

24 
10 
20 
30 
40 
50 

25 

10 
20 
30 
40 
50 

26 
10 
20 
30 
40 
50 

27 
10 
20 
30 
40 
50 

28 
10 
20 
30 
40 
50 

29 O 
10 
20 
30 
40 
50 

30 



Sin, 



3420 



3447 
3475 
3502 
3529 
3556 



03583 



• 3611 

• 3638 

• 3665 

• 3692 
■ 3719 



03746 



• 3773 

• 3800 

• 3827 
3853 
3880 



3907 



• 3934 

• 3961 

• 3987 

• 4014 

• 4041 



4067 



0^4094 
0^4120 
0^4147 
0^4173 
0^4200 



4226 



0^4252 
0-4279 
0-4305 
0-4331 
04357 



d. Tan. 



04383 



0^4410 
0^4436 
0^4462 
0^4488 
0^4514 



4540 



0^4566 
0-4591 
0-4617 
0^4643 
0-46S9 



4694 



0-4720 
0-4746 
0-4771 
0-4797 
04822 



04848 



0-4873 
0-4899 
0-4924 
0-4949 
0-4975 



5000 



Cos. d. 



3639 



3672 
3705 
3739 
3772 
3805 



3838 
3872 
3905 
3939 
3972 
4006 



4040 



4074 
4108 
4142 
4176 
4210 



4244 



4279 
4313 
4348 
4383 
4417 



4452 



4487 
4522 
4557 
4592 

4527 



4663 



4698 
4734 
4770 
4805 
4841 



4877 



4913 
4949 
4986 
5022 
5058 



5095 



• 5132 
5169 
5205 
5242 
5280 



5317 



-5354 
-5392 
-5429 
-5467 
■ 5505 



5543 



5581 
5619 
5657 
5696 
5735 



5773 

Cot. 



Cot. 



2 7475 



7228 
6985 
6746 
6511 
6279 



2 6051 



2-5826 
2-5604 
2-5386 
2-5171 
2-4959 



2 4751 



2-4545 
2-4342 
2-4142 
2 3945 
2-3750 



2 3558 

2-3369 
2-3182 
2-2998 
2-2816 
2-2637 



2. 2460 



2-2285 
2^2113 
2^1943 
2^1775 
2.1609 



2 1445 



2.1283 
2-1123 
2-0965 
2-0809 
2-0655 



2 0503 



2-0352 
2-0204 
2-0057 
1^9911 
1^9768 



1 9626 



1-9486 
1-9347 
1-9210 
1-9074 
1^8940 



1-8807 



1-8676 
1-8546 
1^8417 
1^8290 
18165 



1 8040 



7917 
7795 



1-7675 



7555 
7437 



247 
242 
239 
235 
232 
228 
225 
221 
218 
215 
212 
208 
206 
203 
200 
197 
194 
192 
189 
187 
184 
182 
179 
177 
175 
172 
170 
168 
166 
164 

162 

159 
158 
156 
154 
152 
150 
148 
147 
145 
143 
142 
140 
139 
137 
136 
134 
132 

131 
130 
128 
127 
125 
124 
123 
122 
120 
119 
118 
117 

"d" 



Cos. 



09397 



O 



9336 



9272 



9205 



9135 















9387 
9377 
9366 
9356 
9346 



9325 
9315 
9304 
9293 
9282 



9261 
9250 
9239 
9227 
9216 



9193 
9182 
9170 
9159 
9147 



9123 
9111 
9099 
9087 
9075 



9063 



9050 
9038 
9026 
9013 
9000 
8988 
8975 
8962 
8949 
8936 
8923 



8910 



8897 
8883 
8870 
8856 
8843 



8829 



8816 
8802 
8788 
8774 
8760 



8746 



8732 
8718 
8703 
8689 
8675 



8660 



Sin. 



O 70 

50 
40 
30 
20 
10 

69 
50 
40 
30 
20 
10 

68 
50 
40 
30 
20 
10 

67 
50 
40 
30 
20 
10 

66 
50 
40 
30 
20 
10 

65 

50 
40 
30 
20 
10 

64 
50 
40 
30 
20 
10 

O 63 
50 
40 
30 
20 
10 
62 
50 
40 
30 
20 
10 

61 
50 
40 
30 
20 
10 
O 60 



P. P. 





39 


38 


37 


3€ 


1 


3-9 


3.8 


3-7 


3 


2 


7-8 


7 


6 


7 


4 


7- 


3 


11 


7 


11 


4 


11 


1 


10- 


4 


15 


6 


15 


2 


14 


8 


14- 


5 


19 


5 


19 





18 


5 


18- 


6 


23 


4 


22 


8 


22 


2 


21- 


7 


27 


3 


26 


6 


25 


9 


25- 


8 


31 


2 


30 


4 


29 


6 


28- 


9 


35 


1 


34 


2 


33 


3 


32. 



35 

3-5 



35 

3-5 



34 





27 


27 


26 


2/i 


1 


2-7 


2.7 


2-6 


2- 


2 


5 


5 


5 


4 


5 


2 


5- 


3 


8 


2 


8 


1 


7 


8 


7- 


4 


11 





10 


8 


10 


4 


10- 


5 


13 


7 


13 


5 


13 





12- 


6 


16 


5 


16 


2 


15 


6 


15- 


7 


19 


2 


18 


9 


18 


2 


17- 


8 


22 





21 


6 


20 


8 


20- 


9 


24 


7 


24 


3 


23 


4 


22. 



15 


14 


13 


12 


1.4 


1.4 


1.3 


1- 


2 


9 


2 


8 


2 


6 


2. 


4 


3 


4 


2 


3 


9 


3. 


5 


8 


5 


6 


5 


2 


4. 


7 


2 


7 





6 


5 


6 


8 


7 


8 


4 


7 


8 


7. 


10 


1 


9 


8 


9 


1 


8. 


11 


6 


11 


2 


10 


4 


9. 


13 





12 


6 


11 


7 


10- 



33 

3-3 
6 
9 
2 
5 
8 
1 
4 
7 



11_ 11 

l.lll.l 
2.3 2.2 



10 



3.43.3 
4.64.4 
5.75.5 
6.9 6.6 
8.0 7.7 
9.28.8 
10.319.9 



P. P. 



60''-70'= 



639 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS, 

30°-40° 



30 

10 
20 
30 
40 
50 

310 
10 
20 
30 
40 
50 

33 
10 
20 
30 
40 
50 

33 
10 
20 
30 
40 
50 

34 
10 
20 
30 
40 
50 

35 

10 
20 
30 
40 
50 

36 
10 
20 
30 
40 
50 

37 
10 
20 
30 
40 
50 

38 
10 
20 
30 
40 
50 

39 
10 
20 
30 
40 
50 

40 



Sin. 



5000 






_g 





5150 



5299 



5446 



5592 



5025 
5050 
5075 
5100 
5125 



5175 
5200 
5225 
5250 
5274 



5324 
5348 
5373 
5397 
5422 



5471 
5495 
5519 
5543 
5568 



5616 
5640 
5664 
5688 
5712 



5736 



5759 
5783 
5807 
5830 
5854 



5878 



5901 
5925 
5948 
5971 
5995 
6018 



6041 
6064 
6087 
6110 
6133 



6156 



6i7e 

6202 
6225 
6248 
6270 



6293 



6316 
6338 
6361 
6383 

6405 



6428 



Cos. 



d. 



d. 



Tan. 



5773 

0.5812 
0.5851 
0.5890 
0.5929 
0.5969 



6008 



6048 
6088 
6128 
6168 
6208 



06248 



• 6239 

• 6330 

• 6370 

• 6411 
.3453 



0-6494 

0.6535 
0.6577 
0.6619 
0-6661 
0.6703 



0-6745 



• 6787 
.6830 

• 6873 
.6915 

• 6959 



07002 



0-7045 
0-7089 
0.7133 
0.7177 
0.7221 



0-7265 

0-7310 
0.7354 
0.7399 
0.7444 
0-7490 



O 7535 



0-7581 
0-7627 
0.7673 
0.7719 
0-7766 



7813 

0.7860 
0-7907 
0-7954 
0.8002 
08050 



0-8098 

0-8146 
0-8194 
0-8243 
0-8292 
- 8341 



8391 

Cot. 



39 
39 
39 
39 

39 
3'9 
40 
39 
^0 
^0 
40 
40 
40 
41 
40 
41 
4i 
41 
41 
4l 
42 
42 
42 
42 
42 
42 
43 
42 
43 
43 

43 

43 
44 
44 
44 
44 
44 
44 
45 
45 
45 
45 
45 
46 
46 
46 
46 
47 
47 
47 
47 
47 
48 
48 
48 

48 
49 
49 
49 
49 

1 



Cot. 



17320 



1- 



6643 



6003 



5398 



1- 



482^ 



1. 



7204 
7090 
6976 
6864 
6753 



6533 
6425 
6318 
6212 
6107 



5900 
5798 
5697 
5596 
5497 



5301 
5204 
5108 
5013 
4919 



4733 
4641 
4550 
4460 
4370 



4281 



4193 
4106 
4019 
3933 
3848 



3764 



3680 
3597 
3514 
3432 
3351 



3270 



3190 
3111 
3032 
2954 
2876 



2799 



2723 
2647 
2571 
2497 
2422 



2349 



2276 
2203 
2131 
2059 
1988 



11917 



Tan. 



116 

114 

113 

112 

111 

110 

109 

108 

107 

106 

105 

104 

103 

102 

101 

100 

99 

9'8 

97 

96 

96 

95 

94 

93 

92 

92 

91 

90 

89 

89 

88 

87 
86 
86 
85 
84 
84 
83 
83 
81 
81 
80 

80 

79 
78 
78 
77 
77 
76 
76 
75 
7^ 
74 
73 
73 
73 
19 
7l 
7T 

70 



Cos. 



0-8660 



0.8645 
0.8631 
0.8616 
0.8601 
0-8586 



0-8571 

0-8556 
08541 
0-852'6 
0.8511 
0-8496 



0-8480 



0.8465 
. 8449 
0.8434 
0.8418 
- 8402 



0-8386 
0-8371 
0.8355 
0.8339 
0.8323 
0-8306 



0-8200 



.8274 
.8257 
.8241 
-8225 
-BPOR 



0.8191 



• 8175 
-8158 

• 8141 
-8124 
-8107 



0-8090 
0-8073 
0-8056 
0-8038 
0-8021 
0-8004 



07986 

0-7969 
0.7951 
0-7933 
0-7916 
0-7808 



0-7880 



0-7862 
0-7844 
0-7826 
0-7808 
0-7789 



7771 



-7753 
.7734 
.7716 
.7697 
-7679 



7660 



Sin. 



O 60 

50 
40 
30 
20 
10 

O 59 
50 
40 
30 
20 
10 

58 
50 
40 
30 
20 
10 

5\ 
50 
40 
30 
20 
10 


50 
40 
30 
20 
10 

55 

50 
40 
30 
20 
10 



56 



54 



53 



52 



51 



50 

"o 



p. p. 



49_ 

li 4-9 

2i 9-9 

14-8 

19.8 

24-7 
29-7 
34-6 
39-6 
44-5 



49 48 47 

4.9; 4-8! 4-7 

9-8 9-6[ 9-4 

14-7,14-4 14-1 



19-619-2 
24-5|24-0 
I29.4I28-8 
,34-3133-6 
39-238-4 
I44.1I43.2 



18-8 
23-5 
28-2 
32-9 
37-6 
42-3 



4-5 


4-5 


4-4 


4-3i 


9-1 


9-0 


8-8 


8-6 


13-6 


13-5 


13-2 


12-9 


ll8.2 


18.0 


17-6 


17-2! 


22.7 


22.5 


22-0 


21-5 


27-3 


27.0 


26-4 


25-8 


31-8 


31-5 


30-8 


30-1 


36.4 


36-0 


35.2 


34-4 


40.9 


40-5 


39.6 


38-7 



41 

4-1 

8 
12 
16 
20 
24 
29 
33 
37 



41 

4-1 



40 

4-0 



46 

4-6 
9-2 
13-8 
18-4 
23-0 
27-6 
32-2 
36-8 
41-4 

42 

4-2 
8-4 

12.6 
16-8 
21-0 
25.2 
29.4 
33.6 
37.8 

39 

3-9 



25_ 

2-5 
5.1 
7.6 
10-2 
12.7 
15.3 
17-8 
20.4 
22-9 



25 

2-5 



24 

2-4 



23 

2-3 





22 


22 


18 


1 


1 


2.2 


2-2 


1-8 


1- 


2 


4-5 


4 


4 


3 


7 


3- 


3 


6-7 


6 


6 


5 


5 


5- 


4 


9.0 


8 


8 


7 


4 


7. 


5 


11.2 


11 





9 


2 


9. 


6 


13.5 


13 


2 


11 


1 


10- 


7 


15.7 


15 


4 


12 


9 


12- 


8 


18-0 


17 


6 


14 


8 


14. 


9 


20-2 


19 


8 


16 


6 


16- 



17 


17 


16 


15 


1-7 


1-7 


1.6 


1-5 


3-5 


3-4 


3-2 


3-0 


5-2 


5-1 


4-8 


4.5 


7.0 


6.8 


6.4 


6-0 


8-7 


8.5 


8-0 


7.5 


10-5 


10.2 


9-6 


9.0 


12-2 


11-9 


11-2 


10.5 


14-0 


13.6 


12.8 


12.0 


15-7 


15.3 


14.4 


13.5 



8 
6 

4 
2 

8 
6 
4 
2 

14 

1-4 

2.9 

4-3 

6.8 

7-2 

8-7 

10.1 

11.6 

13-0 



P.P. 



640 



50°-60' 



i| TABLE IX.— NATURAL SINES, COSINES. TANGENTS, AND COTANGENTS. 



40 0. 6428 



Sin, 



0.6450 
0.6472 
0.6494 
0.6516 
0.6538 



0.6560 

0.6582 
0.6604 
0.6626 
0.6648 
8669 

06691 



0.6713 
0.6734 
0.6756 
0.6777 
0.6798 



06820 

0.6841 
0.6862 
0.6883 
0.6904 
0-6925 

6946 



0-6967 
0.6988 
0.7009 
0.7030 
0.7050 



O 7071 



Cos. 



Tan. 



08391 



. 8440 
0-8490 
0-8541 
0-8591 
0-8642 



8693 



0-8744 
0.8795 
0.8847 
0.8899 
0-8951 



09004 



0.9057 
0.9110 
0.9163 
0.9217 
0-9271 



O 9325 



0-9379 
0.9434 
0.9489 
0.9545 
0.9601 



09657 



0-9713 
0.9770 
0.9827 
0.9884 
0-9942 



1 0000 



Cot. 



Cot. 



1.1917 



1.184 

1.177 

1.1708 

1.1640 

1.1571 



1 1503 



1.1436 
1.1369 
1.1303 
1.1237 
1.1171 



11106 



1.1041 
1.0977 
1.0913 
1-0849 
1-0786 



10723 



1-0661 
1.0599 
1.0538 
1.0476 
1-0416 



1 0355 



1.0295 
1.0235 
1 0176 
1.0117 
1-0058 



1 0000 



Tan, 



Cos. 



7660 



7641 
7623 
7604 
7585 
7566 



7547 



• 7528 
.7509 
.7489 
.7470 
.7451 



7431 

• 7412 
.7392 
.7373 
.7353 
.7333 



7313 



7293 
7273 
7253 
7233 
7213 



7193 



7173 
7153 
7132 
7112 
7091 



7071 



Sin. 



O 50 

50 
40 
30 
20 
10 

49 

50 
40 
30 
20 
10 

48 

50 
40 
30 
20 
10 
O 4 

50 
40 
30 
20 
10 

O 46 

50 
40 
30 
20 
10 

O 4^ 



P.P. 



70 


22 


22 


7.0 


2.2 


2.2 


14.0 


4.5 


4.4 


21.0 


6-7 


6.6 


28.0 


9.0 


8.8 


35.0 


11.2 


11.0 


42.0 


13-5 


13-2 


49. 


15.7 


15.4 


56.0 


18.0 


17-6 


63.0 


20.2 


19.8 



69 


21 


20 


6.9 


2-1 


2.0 


13.8 


4.2 


4.1 


20.7 


6-3 


6.1 


27-6 


8-4 


8.2 


34-5 


10-5 


10.2 


41.4 


12.6 


12-3 


48-3 


14.7 


14-3 


55-2 


16.8 


16-4 


62.1 


18.9 


18.4 





68 


19 


19 


18 


1 


6.8 


1.9 


1.9 


1. 


2 


13 


7 


3 .9 


3 


8 


3. 


3 


20 


5 


5.8 


5 


7 


5- 


4 


27 


4 


7.8 


7 


6 


7- 


5 


34 


2 


9.7 


9 


5 


9- 


6 


41 


1 


11.7 


11 


4 


11- 


7 


47 


9 


13.6 


13 


3 


12. 


8 


54 


8 


15-6 


15 


2 


14- 


9 


81 


6 


17-5 


17 


1 


16- 



21 

2.1 

4.3 

6.4 

8.6 

10.7 

12.9 

15.0 

17.2 

19.3 

20 

2.0 

4.0 

6.0 

80 

10-0 

12.0 

14.0 

16.0 

18.0 



P.P. 





68 


67 


66 


65 


61 


64 


63 


62 


61 


60 


59 


59 


58 


58 


1 


6.8 


6-7 


6.6 


6.5 


6.4 


6.4 


6.3 


6.2 


6.1 


6.0 


5.9 


5.9 


5.8 


5.8 


2 


13.6 


13.4 


13.2 


13.1 


12-9 


12.8 


12.6 


12.4 


12.3 


12.1 


11.9 


11.8 


11.7 


11.6 


3 


20.4 


20.1 


19.8 


19.6 


19.3 


19.2 


18.9 


18.6 


18.4 


18.1 


17.8 


17.7 


17.5 


17.4 


4 


27.2 


26.8 


26.4 


26.2 


25.8 


25.6 


25.2 


24.8 


24.6 


24.2 


23.8 


23.6 


23.4 


23.2 


5 


34.0 


33.5 


33.0 


32.7 


32.2 


32.0 


31.5 


31.0 


30.7 


30.2 


29-7 


29.5 


29.2 


29.0 


6 


40.8 


40-2 


39.6 


39.3 


38.7 


38.4 


37.8 


37.2 


36.9 


36.3 


35.7 


35.4 


35.1 


34.8 


7 


47.6 


46-9 


46.2 


45.8 


45.1 


44.8 


44.1 


43.4 


43-0 


42.3 


41.6 


41.3 


40-9 


40.6 


8 


54.4 


53.6 


52.8 


52.4 


51.6 


51.2 


50.4 


49.6 


49.2 


48.4 


47.6 


47.2 


46.8 


46.4 


9 


61.2 


60.3 


59.4 


58.9 


58. D 


57.6 


56.7 


55.8 


55.3 


54.4 


53.5 


53.1 


52.6 


52.2 



57 
5.7 
11.5 
17.2 
23.0 
28.7 
34.5 
40.2 
46.0 
51.7 





57 


56 


56 


55 


53 


54 


53 


53 


52 


52 


5T 


51 


50 


50 


1 


5.7 


5.6 


5.6 


5-5 


5.? 


5.4 


5.3 


5.3 


5.2 


5.2 


5.1 


5.1 


5.0 


5.0 


2 


11.4 


11.3 


11.2 


11-0 


10.9 


10.8 


10.7 


10.6 


10.5 


10.4 


10.3 


10.2 


10. J 


10.0 


3 


17.1 


16.9 


16.8 


16-5 


16.3 


16.2 


16. U 


15.9 


15.7 


15.6 


15.4 


15.3 


15.1 


Ib.O 


4 


22.8 


22.6 


22.4 


22.0 


21.8 


21.6 


21.4 


21.2 


21.0 


20.0 


20.6 


20.4 


20.2 


20.0 


5 


28.5 


28-2 


28.0 


27.5 


27.2 


27.0 


26. V 


26.5 


26.2 


26.0 


25.7 


25.5 


25.2 


25-0 


6 


34.2 


3P.9 


33.6 


33.0 


32.7 
38.1 


32.4 


32.1 


31.8 


31.5 


31.2 


30.9 


30.6 


30.8 


30.0 


7 


39-9 


39.5 


39.2 


38.5 


37.8 


37.4 


37.1 


36.'/ 


36.4 


36.0 


35.7 


35.3 


35-0 


8 45 6 


45.2 


44. 8 


44.0 


43.6 


43.2 


42.8 


42.4 


42.0 


41.6 


41.2 


40.8 


40.4 


40.0 


9 


51.3 


50.8 


50.4 


49.5 


49.0 


48.6 


48.1 


47.7 


47.2 


46.8 


46.3 


45. 9 


45.4 


45.0 



4§ 

4.9 
9.9 
14.8 
19.8 
24-7 
29.7 
34.6 
39. 6 
44. g 



45^-60*' 



641 



•0^ \ 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



Vers. 



00000 



.00000 
•00001 
•00004 
•00007 
•00010 



00015 



•00020 
•00027 
•00034 
-00042 
•00051 



00061 



-00071 
•00083 
•00§95 
•00108 
•00122 



0013' 



•00152 
•00169 
•00186 
•00204 
•00223 



00243 



•00264 
•00286 
•00308 
•00331 
•00355 



003S0 

•00406 
•00433 
•00460 
•00488 
00518 



00548 

•00578 
•00610 
•00643 
•00676 
•00710 



0O745 

•00781 
•00818 
•00855 
•00894 
•00933 



00973 

•01014 
•01056 
.01098 
•01142 
•01186 



01231 



.01277 
.01324 
•01371 
.01420 
•01469 



01519 





1 

2 

3 

3 

4 

5 

6 

7 

8 

8 

10 

10 

ll 

12 

13 

13 

15 

15 

16 
17 
18 
19 
20 
21 
21 
22 
23 
24 
25 

26 

?6 
27 
28 
29 
30 
30 
32 
32 
33 
33 
35 
36 
36 
37 
38 
39 
40 
41 
42 
42 
43 
44 
45 
46 
47 
47 
48 
49 
50 



Vers, d 



Exsec. I d 



00000 



•00000 
.00001 
.00004 
•00007 
.00010 



00015 



•00020 
•00027 
•00034 
•00042 
•00051 



00061 

•00071 
•00083 
00095 
•00108 
•00122 



00137 

•00153 
•00169 
•00187 
•00205 
•00224 



00244 



•00265 
•00286 
•00309 
•00332 
00357 



003 82 



•00408 
•00435 
•00462 
•00491 
•00520 



00551 



•00582 
•00614 
•00647 
•00681 
•00715 



00751 



•00787 
•00824 
•00863 
•00902 
•00942 



00983 



•01024 
•01067 
• OHIO 
•01155 
01200 



01246 

•01293 
•01341 
•01390 
•01440 
•01491 



01542 



Exsec, J d 



Vers. d. Exsec 



01519 



•01570 
•01622 
•01674 
•01728 
•01782 



01837 == 01871 



01893 
01950 
02007 
•02066 
02125 



02185 



•02246 
•02308 
•02370 
•02434 
•02498 



02563 



•02629 
•02695 
•02763 
•02831 
-02900 



•03041 
•03113 
•03185 
•03258 
■03332 



03407 



•03483 
•03559 
•03637 
•03715 
•03794 



•03954 
•04036 
•04118 
•04201 
•04285 



02970 7n 03061 



01542 



01595 
01648 
01703 
01758 
01814 



01929 
01988 
02048 
02109 
02171 



02234 



•02297 
•02362 
•02428 
•02494 
■02562 



02630 



02700 
02770 
02841 
02914 
02987 



•03136 
•03213 
•03290 
•03368 
03447 

03527 



03874 on 04030 



04369 



•04455 
•04541 
•04628 
•04716 
•04805 



04894 



•04984 
•0507| 
•05167 
•05260 
•05354 



05448 



•05543 
•05689 
•05736 
•05833 
•05931 



060,^0 



Vers. 
642 



03609 
03691 
03774 
03858 
03943 



04117 
04205 
04295 
04385 
04476 



04569 



04662 
04757 
04853 
04949 
05047 



05146 



05246 
05347 
05449 
05552 
05656 



05762 



d. 



•05868 
•05976 
•06085 
•06194 
•06305 



06418 



Exsec. d. 



d. 



52 
53 
54 
55 
56 
57 
58 
59 
60 
61 
62 
62 
63 
65 
65 
66 
67 
68 
69 
70 
71 
72 
73 
74 
75 
76 
77 
78 
79 
80 

81 

82 
83 
84 
85 
86 
87 
88 
89 
90 
91 
92 

93 

95 

95 

96 

98 

98 

100 

101 

102 

103 

104 

105 

106 

107 

109 

109 

111 

112 



P. P. 



110 100 90 80 70 



11 
22 
33 
44 
55 
66 
77 
88 
99 

60 

6 
12 
18 
24 
30 
36 
42 
48 
54 

10 

1 
2 
3 
4 
5 
6 
7 
8 
9 



10 
20 
30 
40 
50 
60 
70 
80 
90 

50 

5 
10 
15 
20 
25 
30 
35 
40 
45 



9 
18 
27 
36 
45 
54 
63 
72 
81 

40 

4 
8 
12 
16 
20 
24 
28 
32 
36 



8 


7 


16 


14 


24 


21 


32 


28 


40 


35 


48 


42 


56 


49 


64 


56 


72 


63 



30 

3 
6 
9 
12 
15 
18 
21 
24 
27 



20 

2 

4 

6 

8 

10 

12 

14 

16 

18 



9 9 



3^8 
4^7 
5-7 
66 
7-6 
85 



8 
0^8 



8 



7_ .7 
0^7i07 



1 
2 
3 
3 
4 
5 
8|6 
9l6 



5)1 
2:2 
0;2 
7:3 
5j4 
24 
5 
716 



6 6 5 



H 



4 

0^4 



4 

0^4 



3 

0^3 



2 

0.2 
0.5 
0^7 
1.0 
1.2 



2 

0.2 
0^4 
0^6 



2^2il 



110 

O^IiO^lO-O 



30 
40 
60 
70 
910 
OiO 
2 
3i0 



20.1 
30.1 
40.2 
50-2 
60. 3 
70.3 
8 0.4 
90.4 



P. P. 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
30°-30° 30°-40° 



30 

10 
20 
30 
40 
50 

21 
10 
20 
30 
40 
50 

33 
10 
20 
30 
40 
50 

33 
10 
20 
30 
40 
50 

34 
10 
20 
30 
40 
50 

35 

10 
20 
30 
40 
50 

36 
10 
20 
30 
40 
50 

37 
10 
20 
30 
40 
50 

38 
10 
20 
30 
40 
50 

39 
10 
20 
30 
40 
50 

30 



Vers. d. Exsec 



0603 



.0813 

• 0623 
.0633 

• 0643 
■ 0654 



0664 



• 0674 

• 0685 

• 0896 

• 0706 

• 0717 



0738 



• 0739 

• 0750 
-0761 

• 0772 

• 0783 



0795 



• 0806 

• 0818 

• 0829 

• 0841 
0853 



0864 



• 0876 

• 0888 

• 0900 
.0912 

0924 



0937 



• 0949 

• 0961 
.0974 
■ 0986 

• 0999 



1013 



• 1025 
•1037 
■ 1050 

• 1063 

• 1077 



1090 



1103 
1116 
1130 
1143 
1157 



1170 



• 1184 

• 1198 
.1212 
.1225 

• 1239 



1354 



.1268 

• 1282 

• 1296 

• 1311 

• 1325 



1339 



10 

10 
10 

10 
10 

10 
1(5 
10 

11 

10 

11 

10 

11 
11 
11 
11 
11 
11 
11 
ll 
ll 
ll 

12 
ll 
12 
12 
12 
12 
12 

12 
12 
12 
12 
12 
13 
12 

13 
12 
13 
13 
13 
13 
13 

13 
13 
13 
13 
13 
13 
14 
14 
13 
14 
14 
14 

14 
14 
14 
14 
14 



Vers, i d. 



0643 



• 0653 
.0664 
.0676 
.0688 

• 0699 



0711 



0723 
0735 
0748 
076«J 
0772 



0785 



.0798 
.0811 
.0824 
.0837 
.0855 



0863 



0877 
0890 
0904 
0918 
0932 



0946 



0960 
0975 
0989 
1004 
1019 



1034 



• 1049 
•1064 

• 1079 

• 1094 
-1110 



1136 



.1142 
.1158 
.1174 
.1190 
• 1206 



1333 



.1240 

• 1257 
.1274 
.1291 

• 1308 



1335 



• 1343 
.1361 
.1379 
.1397 

• 1415 



1433 



.1452 
■ 1470 
• 1489 
.1508 
.1527 



1547 



Exsec. 



11 
11 
11 
12 
11 
12 
12 

12 
12 
12 
12 
13 
12 
13 
13 
13 
15 
13 

13 
13 
14 
14 
14 
14 
14 
14 
14 
14 
15 
15 

15 

15 
15 
15 
16 
15 

16 
16 
16 
16 
16 
17 
16 
17 
17 
17 
17 
17 
18 
17 
18 
18 
18 
18 

18 
18 
19 
19 
19 
19 



d. 



30 

10 
20 
30 
40 
50 

31 
10 
20 
30 
40 
50 

33 
10 
20 
30 
40 
50 

33 
10 
20 
30 
40 
50 

34 
10 
20 
3C 
40 
50 

35 

10 
20 
30 
40 
50 

36 
10 
20 
30 
40 
50 

37 
10 
20 
30 
40 
50 

38 
10 
20 
30 
40 
50 

39 
10 
20 
30 
40 
50 

40 



Vers. 



1339 



.1354 
.1369 
.1383 
.1398 
.1413 



1438 



• 1443 
.1458 

• 1473 

• 1489 
■ 1504 



1519 



.1535 
.1550 
.1566 
.1582 
■ 1597 



1613 



.1629 
.1645 
.1861 
.1677 
• 1893 



1709 



• 1726 
.1742 
.1758 
.1775 

• 1792 



1808 



■ 1825 
.1842 
.1859 
.1876 
• 189 3 
1910 



• 1927 
.1944 
.1961 
.1979 

• 1996 



3013 



• 2031 
.2049 

• 2066 
.2084 
■ 2102 



3130 



.2138 
.2156 

• 2174 
.2192 

• 2210 



3338 



2247 
2265 
2284 
2302 
2321 



3339 



Vers. 
643 



± 
15 

14 
14 
15 
15 
15 
15 
15 
15 
15 
15 
15 

15 

15 

15 

16 

15 

16 

15 

16 

1 

16 

16 

16 

16 

16 

16 

16 

17 

16 

16 

17 
17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
18 
12 
17 
18 
18 
18 
18 
18 
18 
18 
18 
18 

18 
18 
18 
18 

18 
J. 



Exsec. 



1547 



.1566 
.1586 
.1606 

• 1626 

• 1646 



1666 



• 1687 
.1707 

• 1728 
.1749 

• 1770 



1793 



.1813 

• 1835 
.1857 
.1879 

• 1901 



1933 



1946 
1969 
1992 
2015 
2038 



3063 

.2086 
.2110 
•2134 
.2158 
.2183 



3307 



■ 2232 

• 2258 

• 2283 

■ 2309 
•2334 



3360 



.2387 
.2413 

• 2440 
.2467 

• 2494 



3531 



• 2549 

• 2576 

■ 2604 

■ 2633 

• 2661 



3690 



2719 
2748 
2778 
2807 
2837 



386: 



2898 
2928 
2959 
2991 
3022 



3054 



19 
19 
20 
20 
20 
20 
20 
20 
21 
21 
21 
21 
21 
21 
22 
22 
22 
22 
23 
22 
23 
23 
23 
23 
24 
24 
24 
24 
24 
24 

25 

25 

25 

25 

25 

26 

26 

26 

26 

27 

27 

27 

27 

27 

28 

28 

28 

28 

29 

29 

29 

29 

30 

30 

30 

30 

31 

31 

31 

31 



Exsec. I d.! 



P. P. 





31 


30 


39 


1 


3.1 


3.0 


2.9' 


2 


6 


2 


6 





5.8! 


3 


9 


3 


9 





8 


71 


4 


12 


4 


12 





11 


61 


5 


15 


5 


15 





14 


5i 


6 


18 


6 


18 





17 


4 


7 


21 


7 


21 





20 


3 


8 


24 


8 


24 





23 


2 


9 


27 


9 


27 





26 


1 



37 

2.7 



4110 
5 13 



9124 



36 

2.6( 



6 20 
3i23 



35 

2-5 



5 

7 

10 

12 

15 

2 17 

8 20 

4 22 



38 

2. J 

5 

8 
11. 
14. 
16. 
19.6 
22-4 
25.2 

34 

2.4 
48 
7.2 
9.6 
12-0 



33 

2.3[ 
4 
6 
9 
11 



1 

2 

3 

4 

5 

6|13 

7II6 

8 18 

9120 



2 

5 11 
8 13 
1 15 
4 17 
7.19 



33 

2.2 
4 
6 
8 



31 



30 

• 1| 2^0 

• 2| 4^0 

• 3i 6.0 
■ 4 80 

• 5llO^O 

• 612.0 

• 7 14-0 

• 8[180 
.9 180 



19 

1-9 
38 

5^7 
7-6 
9.5 

6 11. 4 

7 I3.3JI2 

8 15.2 14 

9 17^lll6 



18 

18; 



10 



15 

1^5| 

3.0! 

4.5| 

6O! 

7^5i 

901 
10. 5| 
12^0:11 
13.5 12 



14 



17 

1-7 



8 10 
6 11 
4 13 
2 15 



16 

16 



13 

1-3 



13 



11 

lfl-1 
2 2 



3!3 
4,4 
5^5 

6 6 

7 7 

8 8 

9 9 



10 

l^G 



2 
3 
4 
5 
6 
7 
8 
9!9 



O 

00 



p. p. 



TABLE X.— NATURAL VERSED SINES ANt) EXTERNAL SECANTS- 
40°-50° 50^-60° 



. 

p. p. 


9 8 7 6 5 4 


0.9 


0.8 


0.7 


0.6 


0.5 


0.4 


1-8 


1.6 


1.4 


1.2 


1.0 


0.8 


2.7 


2-4 


2.1 


1.8 


1.5 


1-2 


3.6 


3-2 


2.8 


2.4 


2.0 


1.6 


4.5 


4.0 


3-5 


3.0 


2-5 


2.0 


5-4 


4.8 


4-2 


3.6 


3-0 


2.4 


6-3 


5.6 


4.9 


4.2 


3.5 


2.8 


7.2 


6.4 


5.6 


4.8 


4.0 


3-2 


8.1 


7.2 


6.3 


5.4 


4.5 


3.6 



Vers. d. 



3339 



2358 
2377 
2396 
2415 
2434 



3453 



2472 
2491 
2510 
2529 
2549 



2568 



2588 
2607 
2627 
2647 
2666 



3686 



2708 
2726 
2746 
2786 
2786 



3806 



2827 
2847 
2867 
2888 

29n« 



3939 



2949 
2970 
2991 
3011 
303'^ 



3053 



3074 
3095 
3116 
3137 
3159 



3180 



■ 3201 

• 3222 

• 3244 

• 3265 

• 3237 



3308 



3330 
3352 
3374 
3395 
3417 



3439 



3481 
3433 
3505 
3527 
3550 



3572 

Vers. 



19 
18 
19 
19 
19 
19 
19 

19 
19 
1? 
19 
19 

19 

19 
19 
20 
19 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 

20 
20 
21 
20 
21 
21 
21 
21 
21 
21 
21 
21 
2l 
21 
21 
21 
21 
21 

22 
21 
22 
21 
22 
22 
22 
22 
22 
22 
22 
22 

T 



Exsec. 



3054 



.3086 
.3118 
.3151 
.3183 
• 3217 



3350 



3284 
3317 
3352 
3386 
3421 



3456 



3491 
3527 
3563 
3599 
3636 



3673 



3710 
3748 
3786 
3824 
3863 



3901 



3941 
3980 
4020 
4060 
4101 



4142 



4183 
4225 
4267 
4309 
4352 



4395 



4439 
4483 
4527 
4572 
4617 



4663 



4708 
4755 
4802 
4849 
4896 



4945 



4993 
5042 
5091 
5141 
5192 



5243 



5294 
5345 
5397 
5450 
5503 



5557 

Exsec. 



32 

32 

32 

32 

33 

33 

34 

33 

34 

34 

34 

35 

35 

38 

33 

38 

37 

37 

37 

37 

38 

38 

39 

38 

39 

39 

40 

40 

40 

41 

41 

4l 
42 
42 
43 
43 
43 
44 
44 
44 
45 
45 
45 
46 
47 
47 
47 
48 
48 
48 
49 
501 
50 
50 
51 
51 
52 
53 
53 
53 

7: 



50 

10 
20 
30 
40 
50 

510 
10 
20 
30 
40 
50 

53 
10 
20 
30 
40 
50 

53 
10 
20 
30 
40 
50 

54 

10 
20 
30 
40 
50 

55 

10 
20 
30 
40 
50 

56 
10 
20 
30 
40 
50 

57 
10 
20 
30 
40 
50 

58 
10 
20 
30 
40 
50 

59 
10 
20 
30 
40 
50 

60 



Vers. 



3573 

.3594 
.3617 
.3639 
.3661 
.3684 



3707 



.3729 
.3752 
.3775 
.3797 
• 3820 



3843 



.3866 
.3889 
.3912 
.3935 
.3958 



d. 



3983 



4005 
4028 
4052 
4075 
4098 



4133 



4145 
4169 
4193 
4216 
4240 



4364 



.4288 
.4312 
.4336 
.4360 
.4384 



4408 



.4432 

• 4456 
.4480 
.4505 

• 4529 



4553 



• 4578 
.4602 

• 4627 
.4651 
.4676 



4701 



4725 
4750 
4775 
.4800 
.4824^ 
4849 



4874 
4899 
4924 
4949 
4975 



5000 

Vers. 



22 

22 

22 

22 

22 

23 

22 

22 

23 

22 

23 

23 

23 

23 

23 

23 

23 

23 

23 

23 

23 

23 

23 

23 

23 

24 

23 

23 

24 

23 

24 

24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
25 
2| 
24 
25 
25 
24 
25 
25 
25 
25 
25 
25 
25 



Exsec. 



5557 



.5611 

• 5666 

• 5721 

• 5777 

• 5833 



5890 



• 5947 

• 6005 
.6064 

• 6123 

• 6182^ 
6343 



• 6303 
.6365 
.6427 

• 6489 

• 6552 



6616 



• 6681 
.6746 
.6811 
.6878 
.6945 



7013 



.7081 
.7150 
.7220 
.7291 
.7362 



■ 7434 



.7507 
.7581 
.7655 
.7730 
.7806 



.7883 



• 7930 
.8039 
.8118 
.8198 
.8279 



8361 



• 8445 

• 8527 

• 8611 

• 8697 

• 8783 



8871 



.8959 
.9048 
.9139 
.9230 
• 9322 



9416 



.9510 
.9606 
.9703 

• 9801 

• 9900 



1 0000 



d. E 



xsec. 



53 

54 

54 

55 

56 

56 

57 

58 

58 

59 

59 

60 

61 

61 

62 

62 

63 

64 

64 

65 

65 

66 

67 

68 

68 

69 

70 

70 

71 

72 

73 

73 
74 
75 
75 
77 
77 
78 
79 
80 
81 
82 
82 
83 
84 
85 
88 
87 
88 
89 
90 
91 
92 
93 
94 
95 
97 
98 
99 
100 



3 

0.3 
0.6 
0.9 
1.2 
1.5 
1.8 
2.1 
2.4 
2.7 



3 

0.2 
0-4 
0^6 
0.8 
1.0 
1.2 
1.4 
1.6 
1.8 



1 
0.1 
0^2 
0^3 
0-4 
0^5 
0^6 
0^7 
0.8 
0.9 



9 

0-9 
1-9 
2.8 
3.8 
4.7 
5.7 
6.6 
7-6, 
8.S 



8 
0-8 
1-7 
2.5 
3.4 
4.2 
5.1 
5.9 
6.8 
7.6 



7 
0.7 
1.5 
2-2 
3-0 
3.7 
4.5 
5.2 
6-0 
6.7 





6 


5 


4 


3 


3 


1 


0.6 


0.5 


0.4 


0.3 


0.2 


2 


1-3 


1.1 


0,9 


0.7 


0.5 


3 


L9 


1.8 


1.3 


i.b 


0.7 


4 
5 


2^6 
3.2 


tl 


2.2 


1:1 


1.0 
1.2 


6 


3.9 


3-3 


2.7 


2.1 


1.5 


7 


4.5 


3.8 


3.1 


2.4 


1.7 


8 


5.2 


4.4 


3.6 


2.8 


2.0 


9 


5^8 


4.9 


4.0 


3^1 


2.2 



35_ 

25 

5.1 

7-6 

10-2 

12^7 



35 

2-5 





5 



5 
8|15.315.0 
7|17.8|17.5 
8 20.4 200 19.6 
9I22.922. 522.0 



34 

2.4' 

4.9 

7^3 

98 

12.2 

14.7 

17^1 





33 


33 


33 


H 


1 


2.3 


2.2 


2.2 


2-1 


2 


4.6 


4.5 


4.4 


4.3 


3 


6.9 


6-7 


6-6 


6.4 


4 


9.2 


90 


8-8 


8.6 


5 


11.5 


11.2 


11^0 


10.7 


6 


13.8 


13.5 


13-2 


12.9 


7 


16.1 


15.7 


15.4 


15.0 


8 


18^4 


18.0 


176 


17-2 


9 


20.7 


20.2 


19.8 


19.3 



30 


30 


19 


19 


2.0 


2.0 


1.9 


1.9 


4.1 


40 


3.9 


38 


6.1 


6-0 


5.8 


5.7 


82 


80 


78 


76 


10.2 


10-0 


9.7 


9.5 


123 


120 


11.7 


11.4 


14-3 


14.0 


13.fe 


13.3 


16.4 


16.0 


15-6 


15.2 


18.4 


180 


17-5 


17^1 



1 

O.I 

0.3 
0.4 

0.6 
0.7 
0.9 
1.0 
1.2 
1.3 

34 33 

2.4 2.3 

4.8 4.7 

7.2 7.0 

9.6 9.4 

12^0 11.7 

14^4!14.1 

16^816.4 

19.21188 

21.6I21.I 



31 

2.1 

4.2 

6.3 

8.4 

10.5 

12.6 

14.7 

168 

18-9 

18 

1-8 

3^7 

5.5 

7.4 

9.2 

ILl 

12.9 

14^8 

166 



P.P. 



644 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS, 
60°-70° 70°-80° 



Ve 



rs. 



5000 



• 5025 
.5050 

• 5076 
.5101 
.5126 



515*^ 



.5177 

.5203 

.5228 

.5254 

• 5279_ 

5305 



.51331 
.5356 

• 5382 

• 5408 

• 5434 



5460 



5486 
5512 
5538 
5564 
5590 



5616 



5642 
5668 
5695 
5721 
5747 



5774 



5800 
5826 
5853 
5879 
5906 



5933 



5959 
5986 
6012 
6039 
6066 



6092 



6115 
6146 
6173 
6200 
6227 



6254 



6281 
6308 
6335 
6362 
6389 



6416 



6443 
6470 
6498 
6525 
6552 



6580 



25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
26 
25 
25 
26 
26 
25 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 

26 
26 
26 
26 
26 
26 
27 
26 
26 
27 
26 
27 
27 
26 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 



Exsec. 



1 0000 



1.0101 
1.0204 
1.0307 
1.0413 
1.0519 



10626 



Vers. I d. 



1.0735 
1.0846 
1.0957 
1.1070 
1.1184 



11300 



1.1418 
1.1536 
1.1657 
1.1778 
1.1902 



12027 



1.2153 
1.2281 
1.2411 
1.2543 
1.2676 



1 2811 



1.2948 
1.3087 
1.3228 
1.3371 
1.3515 



1 3662 



1.381Q 
1-3961 
1.4114 
1.4269 
1.4426 



1 4586 



1.4747 
1.4912 
1.5078 
1.5247 
1.5419 



1 5593 



1.5770 
1.5949 
1-6131 
1.6316 
1-6504 



1-6694 



6888 
7085 
7285 
7488 
7694 



1 7904 



1.8117 
1.8334 
1-8554 
1.8778 
1-9006 



1 9238 



Exsec. d. I ° 



101 
102 
103 
105 
106 
107 
109 
110 
111 
113 
114 
116 
117 
118 
120 
121 
123 
125 
126 
128 
130 
13l 
133 
135 
137 
139 
140 
143 
14} 
146 

148 

151 
152 
155 
157 
159 
161 
164 
166 
169 
171 
174 

177 
179 
183 
184 
188 
190 
194 
196 
200 
203 
206 
210 
213 
216 
220 
224 
228 
23i 



70 

10 
20 
30 
40 
50 

710 
10 
20 
30 
40 
50 

720 
10 
20 
30 
40 
50 

73 Ol 
10 
20 
30 
40 
50 

740 
10 
20 
30 
40 
50 

75 

10 
20 
30 
40 
50 

760 
10 
20 
30 
40 
50 

77 
10 
20 
30 
40 
50 

780 
10 
20 
30 
40 
50 

79 
10 
20 
30 
40 
50 

80 



Vers. d. Exsec 



6580 



.6607 

• 6634 

• 6662 
.6689 
.6717 



6744 



.6772 
.6799 
• 6827 
.6854 
.6882 



6910 



6937 
6965 
6993 
7020 
7048 



7076 



7104 
7182 
7160 
7187 
7215 



7243 



7271 
7299 
7327 
7355 
7383 



7412 



.7440 
.7468 
.7496 
• 7524 
.7552 
.7581 



.7609 
.7637 
• 7665 
.7694 
.7722 



7750 



.7779 
• 7807 
.7835 
.7864 
-7892 
7921 



7949 
7978 
8006 
8035 
8063 



8092 



8120 
8149 
8177 
8206 
8235 



8263 



Vers. 



27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
28 

27 
27 
28 
27 
28 
28 

27 
28 
28 
27 
28 
28 
28 
28 
28 
28 
28 
28 
28 

28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
2? 
28 
28 

I 



1 9238 



1-9473 
1-9713 
1.9957 
2.0205 
2-0458 



2 0715 



2.0977 
2.1244 
2.1515 
2.1792 
2-2073 



2 2360 



2.2653 
2.2951 
2.3255 
2.3565 
2.3881 



2. 4203 



4531 
4867 
5209 
5558 
5915 



2 6279 



2-6651 
2.7031 
2-7420 
2-7816 
2.S9.22 



2-8637 



2-9061 
2.9495 
2.9939 
3-0394 
3-0859 



3 1335 



3-1824 
3-2324 
3-2836 
3-3362 
3-3901 



3 4454 



3-5021 
3-5604 
3-6202 
3.6816 
3 • 7448 



3. 8097 



3-8765 
3-9451 
4-0158 
4-0886 
4-1636 



4. 2408 



4-3205 
4-4026 
4.4874 
4-5749 
4-6653 



4.7587 



Exsec. 



235 

240 

244 

248 

253 

257 

262 

26EJ 

271 

276 

28l 

287 

292 

298 

304 

310 

316 

322 

328 

335 

342 

349 

356 

364 

372 

380 

388 

396 

406 

414 

424 
434 
444 
454 
465 
476 
488 
500 
512 
525 
539 
553 
567 
582 
598 
614 
631 
649 
667 
686 
707 
728 
749 
772 
796 
821 
847 
875 
904 
934 



P. P. 



9 8 7 6 5 






9 





8 





7 





6 


0. 


1 


8 


1 


6 


1 


4 


1 


2 


1 


2 


7 


2 


4 


2 


1 


1 


8 


1- 


3 


6 


3 


2 


2 


8 


2 


4 


2 • 


4 


5 


4 





3 


5 


3 







5 


4 


4 


8 


4 


2 


3 


6 


3- 


6 


3 


5 


6 


4 


9 


4 


2 


3 


7 


2 


6 


4 


5 


6 


4 


8 


4 


8 


1 


7 


2 


6 


3 


5 


4 


4 



4 3 2 






4 





3 


0.20 


1 








8 





6 


0-40 


2 


1- 


1 


2 





9 


0.6'0 


3 


2 


1 


6 


1 


2 


0.8,0 


4 


3. 


2 





1 


5 


1-0.0 


5 


4. 


2 


4 


1 


8 


1-20 


6 


5. 


2 


8 


2 


1 


1-40 


7 


6- 


3 


2 


2 


4 


1-60 


8 


7. 


3 


6 


2 


7 


1-80 


9 


8- 



d. 



8 

0-8 

1-7 

2-5 

3.4 

4.2 

5.1 

715-9 

816-8 

9)7.6 



7_ 
0-7 



6_ 5_ 

0-6 0-5 
1 311-1 
1-9 1-6 
2-6 2-2 
3-2 2.7 
3.9I3.3 
4.5,3.8 
5.214.4 
5.814.9 



4 

0.4 
0.9 
1.3 
18 
2-2 
2-7 
3.1 
3.6 
4.0 



3 

0-3 
0.7 
1-0 
1.4J1 
1.7 1 



2_ 

0-2 

0-5 

0.7 



2 



2 
2 
8'2 



11-5 
41.7 
8 2.0 



9J3-1I2-2 



1 
O-I 
0-3 
0-4 
0-6 





29 


28 


28 


1 


2.9 


2.8 


2.8 


2 


5.8 


5 


7 


5.6 


3 


8.7 


8 


5 


8.4 


4 


11-6 


11 


4 


11.2 


5 


14.5 


14 


2 


14.0 


6 


17.4 


17 


1 


16-8 


7 


20.3 


19 


9 


19-6 


8 


23.2 


22 


8 


22.4 


9 


26-1 


25 


6 


25.2 



27_ 
2-7 



11 



27 
2.7 



26_ 

2.6 



5 13 
2!l5 
9 18 



26 

2.6| 



25 

2 



5 

7 
10 
13.0|12 
15.6115 
18.2^17 



4110 



21-6 21.2,20-8 20 
24-3'23.8i23-4l22 



P. P. 



645 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



80°-85' 



85''-90' 



80 O 

10 
20 
30 
40 
50 

81 

10 
20 
30 
40 
50 

82 
10 
20 
30 
40 
50 

83 

10 
20 
30 
40 
50 

84 

10 
20 
30 
40 
50 

85 O 



Vers. 



8263 



• 8292 

• 8321 

• 8349 

• 8378 

• 8407 



8435 



8608 



8464 
8493 
8522 
8550 
8579 



8637 
8666 
8694 
8723 
8752 



8781 



8810 
8839 
8868 
8897 
8926 



8954 



8983 
9012 
9041 
9070 
9099 



9128 

Vers. 



Exsec. 



_4j 

4 
4 
5 
5 
_5_ 

Ai 

5 
5 
5 
5 
_6 

3 

6 
6 
6 
6 
_l 

JL 

7 
7 
7 
8 
_8 

A: 
8 
9 
9 
9 
10 
10 



7587 



• 8554 

• 9553 

• 058'8 

• 1660 

• 2772 



3924 



• 5121 

• 6363 

• 7654 

• 8998 
.0396 



1853 

• 3372 

• 4957 

• 6613 

• 8344 

• 0156 



2055 



• 4046 

■ 6138 
8336 

• 0651 

■ 3091 



5667 



■ 8391 

■ 1275 
4334 

■ 7585 
^1045 

4737 



Exsec. 



1035 
1072 
1111 
1152 

1196 

1242 
1291 
1343 
1398 
1456 

1519 

1585 
1656 
1731 
1812 
1898 
1991 

2091 
2198 
2315 
244G 
2576 

2723 

2884 
305 
3250 
346C 

3691 



85 O 

10 
20 
30 
40 
50 

86 

10 
20 
30 
40 
50 

87 

10 
20 
30 
40 
50 

88 

10 
20 
30 
40 
50 

89 O 

10 
20 
30 
40 
50 

90 



Vers. 



9128 



■ 9157 

■ 9186 

■ 9215 
• 9244 

■ 9273 



9302 



.9331 
.9360 
.9389 
• 9418 
■ 9447 



9476 



• 9505 

■ 9534 

■ 9564 

■ 9593 

■ 9622 



9651 

.9680 
.9709 
.9738 
• 9767 
.9796 



9825 



• 9854 

■ 9883 

• 9912 
.9942 

■ 9971 



1 0000 

Vers. 



Exsec. 



10 4737 



10 
11 
11 
12 
12 



8683 
2912 
7455 
2347 
7631 



133356 



13 
14 
15 
16 
17 



9579 
6368 
3804 
1984 
1026 



18 1073 



19 
20 
21 
23 
25 



■ 2303 

■ 4937 

• 9256 

• 5621 

■ 4505 



27 6537 



30 
33 
37 
41 
48 



.2576 
-3823 

■ 2015 

■ 9757 
• 1140 



56 2987 



67 
84 

113 
170 
342 



.7573 
.9456 
.5930 
8883 
• 7752 



GC 



Exsec. 





d. 








3946 




4229 




4542 




4892 




5284 




5725 




6223 




6789 




7436 




8180 




9041 




0047 




1230 




2634 




4319 




6365 




8884 


2 


2032 


2 


6039 


3 


1247 


3 


8192 


4 


7741 


6 


1383 


8 


1846 


d. 



P. p. 



29 


,Jt 


2.9 


2-9 , 


5 


9 


5 


8 ' 


8 


8 


8 


7 


11 


8 


11 


6 


14 


7 


14 


5 


17 


7 


17 


4 


20 


6 


20 


3 


23 


6 


23 


2 


26 


5 


26 


1 . 



28 
8 



P. P. 



646 



TABLE XI.— REDUCTION OF BAROMETER READING TO 32"" 


F. 














Inches. 












Temp. 

O 
Fahr. 
























260 


26.5 


27.0 


27.5 


28.0 


28.5 


• 
29-0 


29.5 


30.0 


30.5 


310 


45 


-.039 


-039 


-.040 


-.041 


-.042 


-.042 


-.043 


-.044 


-.045 


-.045 


-.046 


46 


.041 


• 042 


.043 


.043 


• 044 


.045 


.046 


.046 


.047 


.048 


.049 


47 


.043 


•044 


.045 


.046 


•047 


.048 


.048 


.049 


.050 


.051 


•052 


48 


.046 


•047 


•047 


.048 


• 049 


.050 


.051 


.052 


•053 


.053 


.054 


49 


•048 


.049 


.050 


.051 


.052 


.052 


.054 


.054 


• 055 


.056 


.057 


50 


.050 


•051 


.052 


.053 


.054 


.055 


.056 


.057 


• 058 


.059 


.060 


51 


.053 


• 054 


.055 


.056 


•057 


.058 


.059 


.060 


•061 


.062 


• 063 


52 


.055 


•056 


.057 


.058 


•059 


• 060 


.061 


.062 


•064 


.065 


• 066 


53 


.057 


• 058 


.060 


.061 


.062 


.063 


.064 


.065 


•066 


• 067 


• 068 


54 


•060 


.061 


• 062 


•063 


• 064 


.065 


.067 


.068 


.069 


.070 


.071 


55 


• 062 


.063 


.064 


.065 


• 066 


.068 


.069 


.070 


.071 


• 073 


.074 


56 


.064 


.065 


•067 


.068 


.069 


.070 


.072 


•073 


.074 


• 075 


.077 


57 


.067 


.068 


.069 


■070 


•072 


.073 


.075 


.076 


•077 


• 078 


.080 


58 


.069 


.070 


.071 


.073 


.074 


.076 


.077 


.078 


.080 


• 081 


.082 


59 


.072 


.073 


.074 


.075 


.077 


.078 


.080 


•081 


.083 


•084 


.085 


60 


.074 


.076 


.077 


.078 


•079 


.081 


•082 


.084 


.085 


•086 


.088 


61 


•076 


.077 


.079 


.080 


.082 


.083 


.085 


.086 


.088 


• 089 


.091 


62 


.079 


•080 


.082 


.083 


.085 


•086 


.088 


.089 


.091 


•092 


•094 


63 


•081 


.082 


•084 


.085 


.087 


.088 


.090 


.091 


.093 


• 095 


.096 


64 


.083 


.085 


.086 


.088 


.090 


.091 


.093 


.094 


.096 


.097 


.099 


65 


.086 


.087 


.089 


.090 


.092 


.093 


•095 


.097 


.099 


.100 


.102 


66 


.088 


■ 089 


.091 


.093 


.095 


.096 


.098 


.099 


.101 


.103 


• 105 


67 


.090 


.092 


.094 


.095 


•097 


• 099 


.101 


.102 


.104 


.106 


• 108 


68 


.093 


■094 


.096 


.098 


.100 


• 101 


.103 


.105 


.107 


.108 


.110 


69 


.095 


.097 


.099 


.100 


.102 


.104 


.106 


-107 


.110 


.111 


• 113 


70 


.097 


.099 


.101 


.103 


.105 


.106 


.109 


.110 


.112 


.114 


.116 


71 


.100 


.101 


.103 


.105 


.107 


.109 


.111 


.113 


.115 


.117 


.119 


72 


.102 


.104 


.106 


.108 


.110 


.112 


.114 


.116 


.118 


.120 


.122 


73 


.104 


• 106 


.108 


.110 


.112 


.114 


.116 


.118 


.120 


.122 


.124 


74 


• 107 


• 109 


.111 


.113 


.115 


■ 117 


.119 


.121 


.123 


.125 


.127 


75 


.109 


.111 


.113 


.115 


.117 


.119 


.122 


• 124 


.126 


.128 


.130 


76 


.111 


.113 


.116 


.118 


.120 


.122 


.124 


.126 


• 128 


.130 


.133 


77 


• 114 


.116 


.118 


• 120 


.122 


.124 


.127 


.129 


• 131 


.133 


.136 


78 


• 116 


.118 


• 120 


.122 


.125 


.127 


.129 


.131 


.134 


.136 


.138 


79 


.118 


.120 


• 123 


.125 


.127 


• 129 


.132 


.134 


.137 


.139 


.141 


80 


.121 


.123 


.125 


.127 


.130 


.132 


.135 


.137 


.139 


.141 


.144 


81 


.123 


.125 


.128 


.130 


.132 


.134 


.137 


• 139 


• 142 


.144 


.147 


82 


.125 


.128 


• 130 


.132 


.135 


.137 


.140 


• 142 


.145 


.147 


.149 


83 


.128 


.130 


• 133 


.135 


.138 


• 140 


.142 


• 145 


.147 


.149 


.152 


84 


.130 


• 132 


.135 


.138 


.140 


.142 


.145 


• 147 


.150 


.152 


.155 


85 


• 132 


.134 


.137 


.140 


.143 


.145 


.148 


.150 


.153 


.155 


.158 


86 


.135 


• 137 


.140 


.142 


.145 


.148 


.150 


• 153 


.155 


• 158 


.161 


87 


• 137 


.139 


.142 


.144 


.148 


.150 


.153 


.155 


.158 


.161 


• 163 


88 


.139 


.142 


.145 


.147 


• 150 


.152 


.155 


.158 


.161 


.163 


.166 


89 


.142 


.144 


.147 


.150 


■ 153 


.155 


.158 


.161 


.164 


.166 


.169 


90 


.144 


.147 


.150 


.153 


.155 


.158 


.161 


.164 


.168 


.169 


.172 


91 


-.146 


-.149 


-.152 


-.155 


-.158 


-.160 


-.163 


-.166 


-.169 


-.172 


-.175 



647 



TABLE XII.— BAROMETRIC ELEVATIONS.* 



B 



Inches. 

20.0 
20.1 
20.2 
20-3 
20.4 
20.5 
20.6 
20.7 
20.8 
20.9 
21.0 
21.1 
21.2 
21.3 
21.4 
21.5 
21.6 
21.7 
21.8 
21.9 
22.0 
22.1 
22.2 
22.3 
22.4 
22-5 
22.6 
22.7 
22.8 
22.9 
23.0 
23.1 
23.2 
23-3 
23-4 
23.5 
23.6 
23-7 



Feet. 

11.047 
10,911 
10,776 
10,642 
10,508 
10,375 
10,242 
10,110 
9,979 
9,848 
9,718 
9,589 
9,460 
9,332 
9.204 
9,077 
8,951 
8,825 
8,700 
8,575 
8,451 
8,327 
8,204 
8,082 
7,960 
7,838 
7,717 
7,597 
7,477 
7358 
7,239 
7.121 
7,004 
6,887 
6.770 
6,554 
6,538 
6,423 



Diff. for 
• 01. 



Feet. 

■13. 6 
13.5 
13.4 
13.4 
13.3 
13.3 
13.2 
13.1 
13.1 
13. 
12. 9 
12.9 
12.8 
12. 8 
12.7 
12.6 
12.6 
12.5 
12.5 
12-4 
12.4 
12-3 
12.2 
12.2 
12.2 
12.1 
12. 
12.0 
11.9 
11.9 
11.8 
11.7 
11.7 
11.7 
11.6 
11.6 
11.5 



B 



Inches, 

23.7 
23. 8 
23-9 
24-0 
24.1 
24-2 
24-3 
24.4 
24.5 
24.6 
24.7 
24.8 
24.9 
25.0 
25.1 
25-2 
25.3 
25.4 
25.5 
25.6 
25.7 
25.8 
25.9 
26.0 
26.1 
26.2 
26.3 
26.4 
26.5 
26.6 
26.7 
26.8 
26.9 
27.0 
27.1 
27.2 
27.3 
27.4 



Feet. 

6,423 
6,308 
6.194 
6,080 
5,967 
5,854 
5,741 
,629 
,518 
.407 
.296 
,186 
,077 
,968 
4,859 
4,751 
4,643 
4,535 
4,428 
4,321 
4,215 
4.109 
4,004 
3. 899 
3,794 
3 690 
3,586 
3,483 
3.380 
3,277 
3.175 
3,073 
2-972 
2.871 
2 770 
2,670 
2.570 
2.470 



Diff. for 
• 01. 



Feet. 



— 11 


.5 




.4 




4 




.3 




.3 




3 




.2 




1 




.1 




1 







10 


9 


10 


9 


10 


9 


10 


8 


10 


8 


10 


8 


10 


7 


10 


7 


10 


6 


10 


6 


10 


5 


10 


5 


10 


4 


10 


4 


10 


4 


10 


3 


10 


3 


10 


3 


10 


2 


10 


2 


10 


1 


10 


1 


10 


1 


10 





10 





-10 






B 



Inches. 



27 
27 
27 
27 
27 
27 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
31 



Feet. 

2,470 
2,371 
2,272 
2.173 
2,075 
1,977 
1.880 
1,783 
1 686 
1,589 
1,493 
1,397 
1,302 
1.207 



112 
018 
924 
830 
736 
643 
560 
458 
366 
274 
182 
91 

-91 
181 
271 
361 
451 
540 
629 
717 
805 
893 



Diff. for 
.01. 



Feet. 



-9 


9 


9 


9 


9 


9 


9 


8 


9 


8 


9 


7 


9 


7 


9 


7 


9 


7 


9 


6 


9 


6 


9 


5 


9 


5 


9 


5 


9 


4 


9 


4 


9 


4 


9 


4 


9 


3 


9 


3 


9 


2 


9 


2 


9 


2 


9 


2 


9 


1 


9 


1 


9 


1 


9 





9 





9 





9 





8 


9 


8 


9 


8 


8 


8 


8 


-8 


8 



* Compiled from Report of U. S. C. & G. Survey for 1881. App. 10 Table XI. 



TABLE XIII.— COEFFICIENTS FOR CORRECTIONS FOR TEMPERATURE 

AND HUMIDITY.* 



t-vt' 


C 


Diff. for 
1°. 


t + t' 


C 


Diff. for 
1°. 


t + t' 


C 


Diff. for 

1°. 


0° 
10 
20 
30 
40 
50 
60 




1024 
0915 
0806 
0698 
0592 
0486 
0380 


10. 9 
10.9 
10.8 
10. 6 
10.6 
10. 6 


60'' 

70 

80 

90 

100 

110 

120 


+ 
+ 


0380 
0273 
0166 
0058 
0049 
0156 
0262 


10.7 
10.7 
10.8 
10.7 
10.7 
10.6 


120° 

130 

140 

150 

160 

170 

180 


+ 
+ 


0262 
0368 
0472 
0575 
0677 
0779 
0879 


10.6 
10.4 
10.3 
10.2 
10.2 
10.0 



* Compiled from Report of U. S. C. & G. Survey for 1881, App. 10, Tables I, IV. 

648 



TABLE XXX.— USEFUL TRIGONOMETRICAL FORMULA! 





sin a 




1 


1 tan o /l — cos 2a 1 




coseca Vi + tanSa V ^ \^l + cot^ a 



= COS a tan a = Vl — cos- a = 2 sin ^a cos -^a 
1 + cos a 2 tan ^a 



cot ia 1 + tan2 i.^^ 

1 cot a 



= vers a cot ^-a. 
1 



cos a = 



tan a 



seca Vi + cot2 a Vi + tan^a 
= 1 — vers a = sin a cot a = v^l — sin^ a -= 2 cos^ ^a — 1 
= sin a cot ^a — 1 = cos^ ^a — sin^ ^a = 1 — 2 sin^ ^-a, 
1 sin a sec a 1 



cot a cos a cosec a v cosec- a— 1 
= vers 2a cosec 2a = cot a — 2 cot 2a = sin a sec a 
sin 2a 



cot a = 



1 + cos 2a 

1 Ci 



exsec a cot ^a = exsec 2a cot 2a. 
5 a sin 2a 1 + cos 2a 



tan a sm a 1 — cos 2a sin 2a 
= v^cosec^ a — 1 -- cot ^a — cosec a. 
vers a = 1 — cos a = sin a tan ta = 2 sin^ ^a = cos a exsec Co 
exsec a =sec a — 1 = tan a tan ^a = vers a sec a. 

vers a cos +a 



. 1 /vers a sm a 

sm +a = 4 / ;^ — = r- 

y 2 2 cos ia 



sm a 



cos 



1 / 1+ cos a sin a sin a sin ^a 

^a ^^ A/ = = — . 

f 2 2 sin ^a vers a 



tan ^a = vers a cosec a = cosec a — cot a = 



tan a 



XI 1 + COS a 

cot ^a = : = cosec a + cot a ■ 

sm a 

vers+a =1 - V^(i + cos a). 
1 



1 +sec a 
tan a 



exsec a cosec a— cot a 



exsec ^a = 



V^(l + cosa) 



-1. 



649 



TABLE XXX.— USEFUL TRIGONOMETRICAL FORMULA. 



2 tan a 





1 + tan2 a 


cos 2a 


= cos2 a-sin2 a= 1 -2 sm2 a = 2 cos2 a- 1 




1 — tan2 a 




H-tan2a 


tan 2a 


2 tan a 
l-tan2a* 


cot 2a 


= icota-itana-^^*'^-l l-tan2a 



2 cot a 2 tan a 

vers 2a = 2 sin2 a = 1 - cos 2a = 2 sin a cos a tan a. 

exsec 2a=^^^^^ ^_2^an2^ _ 2 sm2 g 

cot a 1 — tan2 a 1 — 2 sin2 a ' 

sin (a ± 6) =sin a cos h ± cos a sin h. 

cos (a ± 6) = cos a cos 6 T sin a sin 6. 

sin a + sin 6 =2 sin i(a + 6) cos i(a-6). 

sin a — sin h =2 sin i(a — 6) cos i(a + 6). 

cos a + cos 6 = 2 cos i(a + 6) cos ^(a — 6). 

cos a — cos 5= —2 sin ^(a + 6) sin ^(a — 6). 



Call the sides of any triangle A, B, C .. and the opposite angles a, h. 
andc. Calls = iU+5 + C). 

tan^(a-6) - . , p tani(a + 6)= . , -, cot ^c. 
A -\-±> A -\- rS 

^ ^cost(a-&) ^smi(a-6) 



.«y^ 



s-5)(s-C) 



sin^a = ^/ ^^ 



/ s{s-A) 



cos^c 

2{s-B){s-C) 

sin & sin c 



versa=- 5C 



Area =Vs(s-A)(s-B)(s-C)=A 



2 sin a 



650 



TABLE XXXI.— USEFUL FORMULA AND CONSTANTS. 



Circumference of a circle (radius = r) = 27rj\ 

Area of a circle = ttv^. 

Area of sector (length of arc = 7) = ^Ir. 

d 

" " '* (angle of arc = a°) = nnr<"'*'^- 

Area of segment (chord = v, mid. ord. = m) = Icm (approx.). 

Area of a circle to radius 1 1 

I 
Circumference of a circle to diameter 1 } = tt =. 3.1415927 

i 
Surface of a sphere to diameter 1 J 

Volume of a sphere to radius 1 = 47r — 3 = 4.1887902 

r degrees = 57.2957795 

I 
Arc equal to radius expressed in -{ minutes = 3437.7467708 

I 
L seconds = 206264.8062471 

Length of arc of 1°, radius unity 0.1745329 

Sine of one second = 0.0000048481 

Cubic inches in United States standard gallon = 231 

Weight of one cubic foot of water at maximum density (therm. 

39°.8 F., barom. 30") 62.379 

Weight of one cubic foot of water at maximum density (therm. 

62° F.) 62.321 

Acceleration due to gravity at latitude of New York in feet per 

square second 32.15945 

Feet in one metre 3.280869 

Metres in one foot 0.304797 

651 



Logarithm. 



0.4971499 

0.622 0886 
1.7581226 
3.536 2739 
5.314 4251 
8.2418774 
4 . 685 5749 
2.363 6120 

1.795 0384 

1.794 6349 

1 . 507 3086 
0.515 9889 
9.4840111 



INDEX. 

Numbers refer to sections except where specifically marked pages (p.). 

Abandonment of existing track 457, c 

Abutments for trestles 142 

Accelerated motion, application of laws to movement of trains 431 

Accidents, danger of, due to curvature," , 418 

Accuracy of earthwork computations . 94 

tunnel surveying 163 

Additional business, methods of securing (or losing) it 455 

train to handle a given traffic, cost of — Table XXVII p. 480 

Adhesion of wheels and rails 333, 334 

Adjustments of dumpy level — Appendix pp. 508, 506 

instruments, general principles — Appendix p. 501 

transit — Appendix pp. 502-506 

wye level — Appendix pp. 506-508 

Advance signals, in block signaling 307 

Advantages of re-location of old lines 456 

tie-plates 244 

Air-brakes 336, 337 

Air resistance — see Atmospheric resistance. 

Allowance for shrinkage of earthwork . 97 

Amorican locomotives, frame 316 

equalizing levers 324 

running gear 323 

system of tunnel excavation 171 

Aneroid barometer, use in reconnoissance leveling 7 

Angle-bars, cost 358, d 

efficiency of 238 

number per mile of track — Table XVII p. 398 

standard 242 

Anglo of slope in earthwork 60 

Apprehension of danger, effect on travel 419, c 

ARCH CULVERTS 191-192 

design 191 

example 192 

Area o2 ©txSverts, computation 178-183 

Ao Co Co E, standard rail sections 226 

iiGsistaut engines — see Pusher engines and Pusher grades. 

Almosphonc resistance, train 342 

653 



654 INDEX. 

Atlantic locomotives, running gear 323 

Austrian system of tunnel excavation 171 

Automatic air-brakes 337 

signaling, track circuit <, « 310 

Averaging end areas, volume of prismoid computed by. ... o 71 

Axles, effect of parallelism 312 

effect of rigid wheels on 311 

radial, possibilities of 313 

size of standard M.C.B „ 332 

Balance of grades for unequal traffic , . 452-454 

determination of relative traffic . . , 454 

general principle 452 

theoretical balance 453 

Balanced grades for one, two, and three engines — Table XXVIII. ... p. 485 

Baldwin Locomotive Works formula for train resistance 348, e 

BALLAST.— Chap. VII. 

cost 200, 358, a 

cross-sections 198 

laying 199 

materials 197 

Banjo signals, in block signaling 308 

Barometer, reduction of readings to 32° F. — Table XI p. 647 

use of aneroid in reconnoissance leveling 7 

Barometric elevations — Table XII p. 648 

coefficients for corrections for temperature and 

humidity— Table XIII p. 648 

Beams, strength of stringers considered as 156 

Bearings, compass, use as check on deflections, 16, 17 

in preliminary surveys 11 

Belgian system of tunnel excavation 171 

Belpaire fire-box. 318 

Blasting 117-123 

use in loosening earth 107, c 

BLOCK SIGNALING.— Chap. XIV. 

Boiler for locomotive , 317, 318 

Boiler-power of locomotives, relation to tractive and cylinder power. . 326 
Bolts — see Track bolts. 

Bonds of railroads, security and profits 369 

Borrow-pits, earthwork 89 

Bowls (or pots) as rail supports 201, 223 

Box-cars, size and capacity 328 

Box culverts 188-190 

old-rail 190 

stone 189 

wooden 188 

Bracing for trestles 140, 141 

design 159 

Brakes — see Train-brakes. 

Brake resistances 346 

Bridge joints (rail) 240 

Bridge spirals ^ 



INDEX. 655 

Bridges and culverts, as affected by changes in alignment 405, 422, 

437, 444, 450 

cost of repairs and renewals 387 

Bridges of standard dimensions for small spans 195 

in block signaling 308 

Bridges, trestles and culverts on railroads, cost 357 

Broken-stone ballast ' 197 

Burnettizing (chloride-of-zinc process) for preserving timber 213 

Capital, railroad, classification of 369 

returns on 369, 370 

Caps (trestle), design 158 

Car mileage, nature and cost — Table XX, p. 425. and 400 

Cars 328-332 

brake-beams 330 

capacity and size 328 

CCiuses of deterioration, items 13 and 14 406 

cost of renewals and repairs 391 

as affected by changes in alignment. . 406, 
423, 437, 444, 450 

draft-gear 331 

gauge of wheels and form of wheel-tread 332 

stresses in car frames 329 

truck frames ; 330 

use of metal 330 

wheels, kinetic energy of 347 

Care and horses, use in earthwork 109, e 

and locomotives, use in earthwork 109, / 

Carte and horses, use in earthwork 109, a 

Cattle r^uards 193 

passes 194 

Center of gravity of side-hill sections, earthwork 92 

Central angle of a curve 21 

Centrifugal force, counteracted by superelevation of outer rail 41, 42 

of connecting-rod, etc., of locomotive 325 

Chairs as supports for double-headed rails 226 

Chemical composition of rails . 232, 233, a 

purification of water 281 

Cinders for ballast ; 197 

Circular lead rails for switches 262 

Clark's formula for train resistance 348, d 

Classification of excavated material 124 

Clearance card in permissive block signaling 304 

spaces in locomotives 321 

Clearing and grubbing for railroads, cost 355 

Coal consumption in locomotives 319 

per car-mile 319 

Columbia locomotives, running gear 323 

Compass, use of, in preliminary surveys 11 

Competitive traffic 409 et seq, 

rates, equality . regardless of distance 410 

Compensation for curvature 427, 428 



656 INDEX. 

Compensation for curvature rate 42§ 

reasons 427 

rules for 428 

Compensators in block signaling 309 

Compound curves 37-40 

modifications of location 39 

nature and use 37 

multiform, used as transition curves 45 

mutual relations of the parts 38 

Comrotmd sections, earthwork 61 

Computation of earthwork 70-95 

approximate, from profiles 95 

using a slide rule 79 

Conducting transportation, cost of . . 393-402 

as affected by changes in curvature 424 

distance 407 

minor grades .... 437 

ruling grades 444 

pusher engines 450 

Coning wheels, effect , 313 

Connecting curve from a curved track to the inside.. 273 

from a curved track to the outside 272 

from a straight track 27 1 

Consolidation locomotives, equalizing levers 324 

frame 316 

running gear 323 

Constants, numerical, in common use — Table XXXI p. 651 

Construction of tunnels 169-174 

Contours, obtained by cross-sectioning 12 

Contractors profit, earthwork 115 

Corbels for trestles 144 

Cost of an additional train to handle a given traffic — Table XXVII . . . 445 

of ballast 200 

of blasting 123 

of chemical treatment of timber 216 

of earthwork : » 106 et seq. 

of framed-timber trestles 150 

of metal ties 222 

of pile trestles 134 

OF RAILROADS.— Chap. XVII. 

detailed estimate 362 

of rails 236 

of ties 209 

of transportation 364 

of treating wooden ties 216 

of tunneling 1 75 

Counterbalancing for locomotives. , 325 

Creosoting for preserving timber. . . ., 212 

Cross-country routes — reconnoissance 4 

Crossings, one straightv one cm"ved track 278 

two curved tracks 279 



INDEX. 657 

Crossings, two curved tracks, numerical example 279 

two straight tracks 277 

Cross-over between two parallel curved tracks, reversed curve 275, b 

curved tracks, straight connecting 

curve 275, a 

straight tracks 274 

Cross-sectioning, for earthwork computations 68 

for preliminary surveys 12 

irregular sections for earthwork computations 87 

Cross-sections of ballast 198 

of tunnels 164 

Cross-ties — see Ties. 

Crown-bars in locomotive fire-box 318 

CULVERTS AND MINOR BRIDGES.— Chap. VI. 

Culverts, arch 191, 192 

► area of waterway — . 178-183 
iron-pipe 186 
old-raH 190 
stone box 189 
tile-pipe 187 

wooden box 188 

CURVATURE.— Chap. XXII. 

compensation for 427, 428 

correction for, in earthwork computations 90-93 

danger of accident due to 418 

effect on cost of conducting transportation 424 

of maintenance of equipment 423 

of maintenance of way 422 

operating expenses of a change of 1 ° — Table XXII , 425 

travel 419 

extremes of sharp 429 

general objections 417 

of existing track, determination 35 

proper rate of compensation 428 

Curve > elements of a 1® 23 

location by deflections ^ 25 

by middle ordinates 29 

by offsets from long chord 30 

by tangential offsets 28 

by two transits 27 

resistance of trains 311 , 312, 345 

effect on cost of conducting transportation . . 424 
maintenance of equipment. . 423 

maintenance of way 422 

Curves, elements of 21 

instrumental work in location 26 

limitations in location o . 34 

method of computing length 20 

modifications of location 33 

mutual relations of elements 22 

obstacles to location 32 



658 INDEX. 

Curves, simple, method of designation 18 

use and value of other methods of location (not using a transit). . 31 
Cylinder power of locomotives, relation to boiler and tractive power. . . 326 

Deflecting rods for operating block signals 309 

Deflections for a spiral 48 

Degree of a curve 18 

Design of culverts 177 et seq, 

framed trestles 151-159 

bracing 159 

caps and sills 158 

floor stringers 156 

posts 157 

nutlocks 253 

pile trestles 133 

tie-plates 245 

track bolts 252 

tunnels 168 

distinctive systems 171 

Development, definition 5 

example, with map 5 

methods of reducing grade > 5 

Disadvantages of re-locations of old lines . . 457 

DISTANCE.— Chap. XXI. 

effect of change on business done 416 

on division of through rates 411 

effect on operating expenses 405- 408 

justification of decrease to save time. 415 

relation to rates and expenses 403 

Distant signals in block signaling 306 

Ditches to drain road-bed . . . , 64 

Dividends actually paid on railroad stock 369 

Double-ender locomotives, running gear 323 

Double-track, distance between centers 62 

Draft gear 331 

* 'continuous" 331 

Drainage of road-bed, value of 64, 65 

Drains in tunnels 168 

Draw -bars 331 

Drilling holes for blasting 118, 119 

Driving-wheels of locomotives 323 

section of 325 

Drop tests for train resistance 350 

Durability of metal ties 219 

rails 234, 235 

wooden ties 204 

Dynamometer tests of train resistance 349 

Earnings of railroads, estimation of 373 

per mile of road 373 

EARTHWORK.— Chap. III. 

Earthwork computations, accuracy 94 

approximate computations from profiles ... 95 



INDEX. 659 

Earthwork computations, relation of actual volume to numerical result 66 

Earthwork, cost 106 et seq., 356 

limit of free haul 105 

method of computing haul 99 ei seq. 

shrinkage 96 

surveys 66-69, a 

Eccentricity of center of gravity of earthwork cross-section 91 

Economics, railroad, justification of methods of computation 367,426 

nature and limitations 365 

of ties 202 

of treated ties 217 

Elements of a 1° curve 23 

simple curve 21 

transition curves — Table IV pp. 520-522 

Embankments, method of formation 98 

usual form of cross-section 58 

Empirical formulae for culvert area 180 

accuracy required , . . . . 183 

value 181 

Engineering, proportionate and actual cos^, in railroad construction . . 353 

Engineering News formula for pile-driving 131 

for train resistance 348, c 

Engineer's duties in locating a railroad 366 

Engine-houses for locomotives 289 

Enginemen, basis of wages 394, 407 

English system of tunnel excavation 171 

Enlargement of tiumel headings 170 

Entrained water in steam 321 

Equalizing-levers on locomotives 324 

Equipment, effect of curvature on maintenance of 423 

Equivalent level sections in earthwork, determination of area 77 

sections in earthwork, determination of area 76 

Estimation of probable volume of traffic and of probable growth 373 

Excavation, usual form of cross-section 58 

Exhaust-steam, effect of back-pressure 321 

Expansion of rails 230 

Explosives, amount used 120 

firing 122 

tamping 121 

use in blasting 117 

Expenditure of money for railroad purposes, general principles 377 

External distance, simple curve 21 

table of, for a 1° curve— Table II , p. 516 

Factors of safety, design of timber trestles 155 

Failures of rail joints 241 

Fastenings for metal cross-ties 221 

Field work for locating a simple curve 26 

a spiral 50 

Fire-box of locomotive 318 

required area 318 

Fire-brick arches in locomotive fire-box 318 



660 INDEX. 

Fire protection on trestles 148 

Five-level sections in earthwork, computation of area 80 

Fixed charges, nature and ratio to total disbursements 378 

Flanges of wheels, form 332 

Flanging locomotive driving-wheels, effect 314 

Floor systems for trestling 143-150 

Formation of embankments, earthwork 96-98 

railroad corporations, method 368 

Formulae for pile-driving 131 

required area of culverts 180 

train resistance 348 

trigonometrical — Table XXX pp. 649, 650 

useful, and constants — Table XXXI p. 651 

Fouling point of a siding , 310 

Foundations for framed trestles 139 

FRAMED TRESTLES 135-159 

abutments 142 

bracing 140, 141 

cost 150 

design 135, 151-159 

foundations 139 

joints o . . . 136 

multiple stony construction 137 

span . 138 

Frame of locomotive, construction 316 

Free haul of earthwork, limit of 105 

Freight yards 295-299 

general principles 295 

minor yards . 297 

relation of yard to main tracks ,. = „,,, = .. 296 

track scales ,,,,,. o ., o 299 

transfer cranes » c . . . . . » , o . o . , . , , 298 

French system of tunnel excavation 171 

Friction, laws of, as applied to braking trains , 334 

Frogs, diagrammatic design, ,.„... ,o, » 255 

EUiot, illustrated in Fiate VIII, opposite po 267. 

for switches , « 255, 256 

to find frog number . 256 

trigonometrical functions — Table III, j)= 619. 
Weir, illustrated in Plate Vlil, opposite p. 267. 

Fuel for locomotives, cost of 395, 407, item 22, and Table XX 

as affected by changes in alignment, 407, 424, 

437, 444, 450 

German system of timnel excavation 171 

GRADE— Chap. XXIII. 

(see Minor grades. Pusher grades. Ruling grades). 

accelerated motion of trains on , 431 

distinction between ruling and minor grades 430 

in tunnels 165 

line, change in, based on mass diagram o 104 

resistance of 344 



INDEX. 661 

Grade, undulatory, advantages, 'disadvantages, and safe limits 434 

virtual 432 

use, value, and misuse 433 

Grade resistance of trains 344 

Gravel ballast 197 

Gravity tests of train resistance 350 

Grate area of locomotives 318, 320 

ratio to total heating surface 320 

Gravity, effect on trains on grades 344 

tests of train resistance 350 

Ground levers for switches 259 

Growth of railroad traffic 373 

affected by increase of facilities 375 

Guard rails for smtches . . . . , 261 

lor trestles 145 

Guides around curves and angles (signaling mechanism) 309 

Gun-powder pile-drivers 130 

Hand-brakes. 335 

Haul of earthwork, computation of length , 99 et seq. 

cost 109, 116 

limit of profitable 116 

method depending on distance hauled 110 

Headings in tunnels 169 

Heating surface in locomotives 320 

Hire of equipment (rolling stock) , nature of item and cost — Table XX . . 400 

Hoosac Tunnel, surveys for 160, 163 

Hump in a grade, operating value of removal 439 

I-beam bridges, standard 195 

IMPROVEMENT OF OLD LINES.— Chap. XXIV. 

classification 455 

Inertia resistances 347 

Instrumental work in locating simple curves 26 

spirals 50 

Interest on cost of railroads during construction 360 

Iron pipe culverts 186 

Irregular prismoids, volume 82 

sections in earthwork, computation of area 81 

Joints, framed trestles 136 

rail 237-243 

Journal friction of axles 343, b 

Kinetic energy of trains 431 

Kyanizing (bichloride-of-mercury or corrosive-sublimate process) for 

preserving timber 214 

Land and land damages, cost 354 

Lateral bracing for trestles 141 

Length of rails. . 229 

a simple curve 20 

a spiral 53 

Level, dumpy, adjustments of — Appendix p. 508 

wye, adjustments of — Appendix p. 606 

Leveling, location surveys 16 



662 INDEX. 

Level sections, volume of prismoids surveyed as 74 

numerical example 75 

JLife of locomotives 327 

Limitations in location of track 34 

of maximum curvature 429 

Lining of tunnels 166 

Loading earthwork, cost 108 

of trestles 154 

Local traflSc, definition and distinction from through 409 

Location of stations at distance from business centers, effect 376 

Location Surveys — paper location 15 

surveying methods 16 

Locomotives, as affected by changes in alignment .... 406, 423, 437, 444, 450 

causes of deterioration, item 12 406 

cost of renewals and repairs 390 

general structiu-e 316-326 

life of 327 

types permissible on sharp curvature 420, b 

(For details, look for the particular item.) 

Logarithmic sines and tangents of small angles — Table VI p. 543 

sines, cosines, tangents, and cotangents — Table VII. . . p. 546 

versed sines and external secants — Table VIII p. 591 

Logarithms of numbers — Table V p. 523 

Long chords for a 1° curve — Table II p. 516 

of a simple curve 21 

Longitudinal bracing of a trestle 140 

Longitudinals (rails) 201, 224 

Loop — see Spiral. 

Loosening earthwork, cost 107 

Loss in traffic due to lack of facilities 376 

Magnitude of railroad business 363 

Maintenance of equipment, as affected by changes in curvature 423 

distance 406 

minor grades. . . . 437 

ruling grades. . . . 444 

pusher-engines 450 

cost of 389-392 

Maintenance of way as affected by changes in curvature 422; 

distance. 405 

minor grades 437 

ruling grades 444 

pusher-engines 450 

cost of 384-388 

Maps, use of, in reconnoissance S 

Mass curve, area 102 

properties 101 

diagram, effect of change of grade line 104 

haul of earthwork 100 

value 103 

Mathematical design of switches 262-275 

Measurements, location surveys, ,.,,,,, t .... f f r « f 1 1 « w 1 1 1 < * t * » t • » 16 



INDEX. 663 

Mechanism of brakes 335-337 

METAL TIES 218-223 

cost 222 

durability 219 

extent of use 218 

fastenings 221 

form and dimensions 220 

Middle areas, volume of prismoid computed from 72 

ordinate of a simple curve 21 

Mileage, car 400 

locomotives, average annual 327 

MINOR GRADES 435-439 

basis of cost 435 

classification 436 

effect on operating expenses 437 

estimate of cost of one foot of change of elevation. . . . 438 
operating value of the removal of a hump in a 

grade 439 

Minor openings in road-bed 193-195 

Minor stations, rooms required, construction 287 

MISCELLANEOUS STRUCTURES AND BUILDINGS.— Chap. XII. 

Modifications in location, compound curves 39 

simple curves 33 

Mogul locomotives, running gear 323 

Monopoly, extent to which a railroad may be such 371 

Mountain routes — reconnoissance 5 

*• Mud " ballast 197 

sills, trestle foundations 139, 6 

Multiform compound curves used as spirals 45 

Multiple story construction for trestles 137 

Myer's formula for culvert area 180 

Natural sines, cosines, tangents, and cotangents — Table IX p. 637 

versed sines and external secants — Table X p. 642 

Non-competitive traffic, definition 409 

effect of variations in distance 413, 414 

extent of monopoly 371 

Notes — ^form for cross-sectioning 12 

location surveys 17 

reconnoissance 7 

Number of a frog, to find 256 

of trains per day, probable 374 

Nut-locks, design 253 

Obstacles to location of trackwork 32 

Obstructed curve, in curve location 32, c 

Old lines, improvement of — Chap. XXIV 

rail culverts 190 

Open cuts vs. tunnels 174 

OPERATING EXPENSES.— Chap. XX. 

detailed classification— Table XX 382-402 

effect of change of grade — 439, and Table 

XXIV p. 472 



664 INDEX. 

operating expenses, effect of curvature on 421-426 

distance on 405-408 

estimated cost of each additional foot — Table 

XXI 408 

of each additional mile — Table 

XXI 408 

of 1° additional of central angle 

— Table XXII 425 

fourfold distribution 379 

per train mile 380 

reasons for uniformity per train mile 381 

(For details look for the particular item.) 

Operation of trains, effect of curvature on 420 

Oscillatory and concussive velocity resistances, train 342 

Ordinates of a spiral 47 

Paper locatiou*in location surveys 15 

Passengers carried one mile , 363 

Physical tests of steel splice bars 243, a 

steel rails 233, a 

Ptrks, use in loosening earth 107, b 

Pile bents 129, 133 

driving. 130 

driving formulae 131 

points and shoes 132 

trestles, cost 134 

design 133 

PILE TRESTLES 129-134 

pilot truck of locomotive, action 315 

PIPE CULVERTS 184-187 

> 

advantages 184 

construction 185 

iron 186 

tile 187 

Pipe compensator 309 

Pipes, use in block signaling 309 

Pit cattle guards 193 

Platforms, station 286 

Ploughs, use in loosening earth 107, a 

Point of curve 21 

inaccessible, in curve location 32, b 

Point of tangency 21 

inaccessible, in curve location 32, b 

Point-rails of switches, construction 258 

Point-switches 258 

Pony truck of locomotive, action 315 

Portals, tunnels, methods of excavation 173 

Posts, trestle, design of 157 

Preliminary financiering of railroads — Chap. XIX.. and 352 

Preliminary surveys — cross-section method 11 

"first" and "second". 14 

general character 10 



INDEX. 665 

Preliminary surveys, value of re-surveys at critical points 14 

Preservative processes for timber, cost 216 

general principle 210 

methods 211-215,217 

Prismoidol correction for irregular prismoius, approximate value 85 

true value 83 

in earthwork computations, comparison of exact 

and approximate methods . . 86 

formula, proof 70 

Prismoid, irregular, computation of volume 82 

Prismoids, in earthwork computations 67 

Profit and loss, dependence on business done 372 

small margin between them for railroad promoters 370 

Profits (and security) in the two general classes of railroad obligations. . . 369 

Profit, in earthwork operations 115 

PROMOTION OF RAILROAD PROJECTS.— Chap. XIX. 

Pumping, for locomotive water-tanks 283, 284 

Pusher grades 446-451 

comparative cost 450, 451 

general principles 446 

required balance between throug'i and pusher grades. . 447 

required length 449 

Pusher engines, cost per mile — Table XXIX p. 488 

operation 418 

service 450 

Radiation from locomotives 321 

into the exhaust-steam, 321 

Radii of curves — Table I p. 512 

Radius cf curvature (of track), relation to operating exi>en3es 421 

Rail braces 244 

expansion, resistance at joints and ties to frea expansion 251 

FASTENINGS.— Chap. X. 

gap. effect of, at joints 239 

joints 237-243, a 

"Bonzano" 243 

*'Cloud" 243 

"Continuous" 243 

*'Fisher" 243 

"Weber" 243 

"100 per-eent" 243 

"bridge" 240 

effect of rail gap 239 

efficiency of angle-bar 238 

failures 241 

later designs. 243 

specifications 243, a 

"supported" 238,240 

"suspended" 238, 240 

theoretical requirements for perfect 237 

sections 225, 226 

A.S.C.E 226 



666 INDEX. 

Rail sections, "bridge" 225 

"bull-headed'' 225, 226 

compound 240 

**pear" section 225 

radius of upper corner, effect 226 

reversible 226 

"Stevens" 225 

" Vignoles" 225 

RAILS.— Chap. IX. 

branding 233-a,ll 

cast-iron 225 

cost 236, 358, c 

of, as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher engines 450 

of renewals of 234 235, 385 

chemical composition 232, 233, a 

effect of stiffness on traction 228 

expansion 230 

stresses caused by prevention of expansion 230 

rules for allowing for 231 

inspection 233-a-12 

length 229 

allowable variation 233, a, 8 

45- and 60-foot rails 229 

No. 2 233-a-13 

physical properties 233 a, 3 

relation of weight, strength, and stiffness 228 

temperature when exposed to sun 231 

testing 233, 233, a 

tons per mile 358, c 

wear on curves 235 

tangents 234 

weight, allowable variation 233-a-7 

for various kinds of traffic 358 

Rates based on distance, reasons 404 

through, method of division of 410 

Receipts (railroad), effect of distance on 409-416 

Reconnoissance over a cross-country route 4 

surveying, leveling methods 7 

surveys 1-9 

character of 1 

distance measurements 8 

mountain route 5 

selection of general route 2 

value of high grade work 9 

through a river valley 3 

Reduction of barometer reading to 32° F.— Table XI p. 647 

Renewal of rails, cost of 234, 235, 385 



INDEX. 667 

Renewal of rails, cost of, as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher engines 450 

Renewal of tie* cost of 208, et seq. 386 

as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher engines 450 

regulations governing it 208 

Repairs and renewals of locomotives, cost 390 

as affected by change of dis- 
tance 406 

curvature 423 

minor grades.. 437 
by ruling grades 444 

Repairs of roadway, cost or 384 

as affected by changes in curvature 422 

distance 405 

minor grades. . . . 437 
ruling grades .... 444 

pusher engines 450 

Repairs, wear, depreciation, and interest on cost of plant ; cost for earth- 
work operations 113 

Replacement of a compound curve by a curve with spirals 53 

simple curve by a curve with spirals 51 

Requirements, nut-locks 251 

perfect rail-joint 237 

spikes 247 

track-bolts 251 

Resistances internal to the locomotive 341 

(see Train Resistance.) 

Revenue, gross, distribution of 378 

Roadbed, form of subgrade 63 

width for single and double track 62 

Roadway, cost of repairs of 384 

as affected by changes in ciu-vature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher engines 450 

Roadways, earthwork operations, cost of keeping in order 112 

Rock ballast 197 

Rock cuts, compound sections 61 , 62 

RolHng friction of wheels 343, a 

ROLLING STOCK.— Chap. XIV. 

Rotative kinetic energy of wheels of train 347, 431 

Rules for switch-laying. 276 

Ruling grades ..,,.,,.., 440-445 



668 INDEX. ^ V 

Ruling grades, choice of 441 

definition 440 

operating value of a reduction in rate of 445 

proportion of traffic affected by 443 

Run-off for elevated outer rail 43 

Running gear of locomotives, types 323 

Scales, track 298 

Scrapers, use in earthwork 109, d 

Screws and bolts, as rail-fastenings 249 

Searles*s formula for train resistance 348, a 

Section-houses, value, construction 288 

Selection of a general route for a railroad 2 

Semaphore boards, in block signaling 308 

Setting tie-plates, methods 246 

Shafts, tunnel, design 167 

surveying 161 

Shells and small coal, used as ballast 197 

Shifting centers for locomotive pilot trucks, action 315 

Shoveling (hand) of earthwork, cost 108, a 

(steam) of earthwork, cost 108, h 

Shrinkage of earthwork 96 

allowance 97 

Side-hill work, in earthwork computations 88 

correction for curvature 91 

Signaling, block, "absolute '^ blocking 304 

automatic 305 

manual systems 302-304 

permissive 304 

Signals, mechanical details 308 

Sills for trestles, design 158 

Simple curves 18-36 

Skidding of wheels on rails 333, 334 

Slag, used for ballast ..*..... 197 

Slide-rule, in earthwork computations 79 

Slipping of wheels on rails, lateral 312 

longitudinal 311 

Slopes in earthwork, for cut and filL 60, 62 

effect and value of sodding. ........ 65 

Slope-stake rod, automatic * 69, a 

Slope-stakes, determination of position 69 

Sodding slopes, effect and value 65 

Spacing of ties 206 

Span of trestles 138 

Specifications for earthwork 125 

steel rails 233, a 

steel splice-bars 243, a 

wooden ties. 207 

Speed of trains, reduction due to curvature 419, a 

relation to superelevation of outer rail , . 41 , 42 

relation to tractive adhesion , , , , . 334, e 

Spikes, , .,,. .247-250 



INDEX. 669 

spikes, cost 358, d 

driving 248 

number per mile of track 358 , d 

requirements in design 247 

"wooden," for plugging spike-holes 250 

Spirals, bridge and tunnel 5 

(see Transition Curves.) 
Splice-bars — (see Angle-bars). 

Split stringers, caps, and sills 129, 143 

Spreading earthwork, cost Ill 

Stadia method — for preliminary surveys 13 

Stand pipes, for locomotive water-supply , 285 

Starting grade at stations, reduction of 460 

Staybolts for locomotive fire-boxes 318 

Stays, in locomotive fire-box 318 

Steam pile-drivers 130 

Steam-shoveling of earthwork 108, h 

Stiffness of rails, effect on traction 228 

Stocks of railroads, security and profits 369 

Stone box culverts 189 

foundations for framed trestles 139, c 

Straight connecting curve between two parallel curved tracks 275, a 

from a curved main track 273 

frog rails, effect on design of switch 263 

point rails, effect on design of switch 264 

Strength of timber 153 

factors of safety 155 

required elements for trestles 152 

Stringer bridges, standard, steel 195 

Stringers, design 156 

for trestle floors 143 

Stub-switches , 257 

Subchord, length 19 

Subgrade, of roadbed, form ,. , . . 63 

Superelevation of the outer rail on curves, L. V. R. R. rim-off ....... 43 

on trestles ...,,, 147 

practical rules , , , 42 

standard on N. Y. N. H, 

& H. R. R 42 

theory , , 41 

Superintendance, cost in earth operations ,..,., 114 

of conducting transportation , 393 

of maintenance of equipment ,.,...., 389 

Supported rail-joints ♦.,... 240 

Surface cattle guards ♦ . . , 193, h 

surveys for tunneling , , . 160 

Surveys and engineering expenses for railroads, cost , 353 

accuracy 163 

for tunneling 160-163 

with compass 11 

Suspended rail-joints 240 



670 INDEX. 



Swinging pilot truck on locomotive 315 

Switchbacks 5 

Switch construction 254 261 

essential elements 254 

frogs 255, 256 

guard rails 261 

point 258 

stands 259 

stub 257 

tie rods 260 

SWITCHES AND CROSSINGS,. .Chap XI 

Switches, mathematical design 262-276 

comparison of methods 266 

using circular lead rails 262 

using straight frog rails 263 

using straight point rails 264 

using straight frog rails and straight 

point rails 265 

Switching engines, running gear 323 

used in pusher-engine service 448 

Switch leads and distances — Table III p. 519 

laying, practical rules 259 

stands 276 

TABLES. numbers refer to pages, not sections. 

I. Radii of curves 512-515 

II. Tangents, external distances and long chords for a 1° curve, 

516-518 

III. Switch leads and distances 519 

IV. Elements of transition curves 520-522 

V. Logarithms of numbers 523-542 

VI. Logarithmic sines and tangents of small angles 543-545 

VII. Logarithmic sines, cosines, tangents, and cotangents. .. 546-590 

VIII. Logarithmic versed sines and external secants 591-636 

IX. Natural sines, cosines, tangents, and cotangents 637-641 

X. Natural versed sines and external secants 642-646 

XL Reduction of barometer reading to 32° F 647 

XII. Barometric elevations 648 

XIII. Coefficients for corrections for temperature and humidity. . 648 

XIV. Capacity of cylindrical water-tanks in United States standard 

gallons of 231 cubic inches 301 

XV. Number of cross- ties per mile 396 

XVI. Tons per mile (with cost) of rails of various weights 397 

XVII. Splice bars and bolts per mile of track 398 

XVIII. Railroad spikes 398 

XIX. Track bolts, average number in a keg of 200 pounds 398 

XX. Classification of operating expenses of all railroads, to face p. 425 

XXI. Effect on operating expenses of changes in distance 439 

XXII. Effect on operating expenses of changes in curvature 453 

XXIII. Velocity head of trains 463 

XXIV. Effect on operating expenses of changes in grade 472 

XXV. Tractive power of locomotives 475 



INDEX. 



671 



TABLES. 

XXVI. Total train resistance per ton on various grades 477 

XXVII. Cost of an additional train to handle a given traffic 480 

XXVIII. Balanced grades for one, two, and three engines 485 

XXIX. Cost per mile of a pusher-engine 488 

XXX. Useful trigonometrical formulae 649, 650 

XXXI. Useful formulae and constants. 651 

Talbot's formula for culvert area 180 

Tamping for blasting 121 

Tangents for a 1° curve — Table II p. 516 

Tangent distance, simple curve 21 

Tanks, water, for locomotives 282 

capacity of cylindrical tanks 282 

track 284 

Temperature allowances, while laying rails 231 

Ten-wheel locomotives, running gear 323 

Telegraph lines for railroads, cost 361 

TERMINALS.— Chap. XIII. 

inconvenient, resulting loss. . 376 

justification for great expenditures 376 

Terminal pyramids and wedges, in earthwork 59 

Tests for splice bars 243, a 

for rails 233, a 

to measure the efficiency of brakes 338 

Three-4evel sections in earthwork, determination of area 78 

* numerical example 78 

Throw of a switch 262 

Through traffic, definition 409 

division of receipts between roads 410 

effect of changes in distance on receipts 411 

Tie-plates 244-246 

advantages 244 

elements of design . 245 

method of setting 246 

Tie rods, for switches 260 

TIES.— Chap. VIII. 

cost of renewal of 208 et seq. 386 

as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher-engines 450 

metal 218-223 

cost 222 

durability 219 

extent of use 218 

fastenings 221 

form and dimensions 220 

number per mile of track — Table XV p. 396 

on trestles 146 

wooden 203-217 



672 UNTBEX. 

Ties, wooden, choice ol wood 203; 

con&tructioB 207 

cost 209, 358, b 

dimensions, 205 

durability 20 1 

economics 202 

quality of timber 207 

spacing 206 

specifications 207 

Tile drains, to drain roadbed G i 

pipe culverts 187 

Timber, ehoice for trestles 14^ 

piles 129 

ties 203 

strength of 153 

Tons carried one mile 363 

Topographical maps, use of, in reconnaissance '6 

Track bolts, average number in a keg of 200 pounds 358, d 

cost 358, d 

design 252 

essential requirements. 251 

number required per mile 358, d 

circuit for automatic signaling 310 

laying on railroads, cost 358 , e 

scales 299 

Tractive power of locom.otives. Table XXV p. 475 and 322 

relation to boiler and cylinder j^ower. .326 

Traffic, classification of ' 409 

TRAIN-BRAKES 333-339 

automatic 337 

brake-shoes 339 

general principles 333, 334 

hand-brakes 335 

straight airbrakes 338 

tests for efficiency. . 338 

Train length limited by curvature. . . » - 420, a 

load, financial value of increasing 444 

maximum on any grade 442 

loads, methods of increasing 455, &, 458 et seq. 

RESISTANCE.— Chap. XVI. 

total, per ton, on various grades — Table 

XXVI p. 477 and 444 

service, cost of, 397 and Table XX 

as affected by changes in alignment, 407, 424, 437, 

444, 450 
supplies and expenses, cost of, in conducting transportation — ■ 

398 and Table XX. 
wages — (see Train service). 

Transfer cranes in freight yards 298 

Transit, adjustments of P-p. 501-506 

Transition curves 41-53 



INDEX. 673 

Transition curves, Table IV PP. 520-522 

application to caioapound curves 52 

field work 50 

fundamental principle 44 

replacing a compound curve by curves with spirals 53 
simple curve by a curve with spirals. ... 51 

required length 46 

their relation to tangents and simple curves. 49 

to find the deflections from any point 48 

ordinates 47 

Transportation, effect of curvature on conducting 424 

TRESTLES.— Chap. IV. 

cost 150 

extent of use •• 126 

framed * 135-150 

pile 129-134 

posts, design 157 

required elements of strength 152 

sills, design . . . , 158 

stringers, design 156 

timber 149, 153 

vs. embankments. 127 

Trucks, car 330 

four-wheeled, action on curves 312 

locomotive pilot ., 315 

with shifting center 315 

TUNNELS.— Chap. V. 

cost 175 

vs. open cuts 174 

Tunnel cross-sections 164 

design , 164-168 

drains 168 

enlargement 170 

grade 165 

headings /. 169 

lining 166 

portals 173 

shafts 161, 167 

spirals ^ . . * 5 

Turnout, connecting curve from a straight track 271 

from a curved track to the outside 272 

to the INSIDE. 273 

double, from straight track 260 

dimensions, development of approximate rule for above. . .. 267 

from INNER side of curved track 268 

from OUTER side of curved track 267 

Turnouts with straight point rails and straight frog rails, dimensions of 

—Table III p. 519 

two, on same side 270 

Turntables for locomotives 292 

Two-level ground, volume of prismoid surveyed as 73 



674 INDEX. 

Underground surveys in tunnels 162 

Undulatory grades, advantages, disadvantages, and safe limits 434 

Unit chord, simple curves 18 

Upright switch-stands. 259 

Useful formulae and constants — Table XXXI p. 651 

trigonometrical formulae — Table XXX p. 647, 648 

Valley route — reconnoissance 3 

Velocity head applied to theory of motion of trains 431 

as applied to determination of train resistance 350 

of trains— Table XXIII p. 463 

Velocity of trains, method of obtaining 458 

resistances, train 342 

Ventilation of a tunnel during construction 172 

Vertex inaccessible, curve location 32, a 

of a curve 21 

Vertical curves, mathematical form 56 

necessity for use 54 

numerical example. 57 

required length 55 

Virtual grade, reduction of 458-460 

profile, construction of 432 

use, value, and possible misuse 433 

Vulcanizing, for preserving wooden ties 211 

Wages of engine- and roundhouse-men 394; 407, item 21 

as affected by changes in align- 
ment. .407, 424, 437, 444, 450 

Wagons, use in hauling earthwork 109, b 

Water for locomotives, chemical qualities 281 

consumption and cost 396; 407, item 23 

methods of purification 281 

stations and water supply 280-285 

location 280 

pumping 283 

required qualities of water 281 

stand-pipes 285 

tanks 282 

track tanks 284 

table in locomotive fire-box 318 

tanks for locomotives 282 

capacity of cylindrical tanks 282 

protection from freezing 282 

way for culverts 178-183 

Watering stock 369 

Wear of rails on curves 235 

on tangents 234 

Weight of rails, 226, 227, and Table XVI P 397 

Wellhouse (zinc-tannin) process for preserving timber 215 

Wellington's formula for train resistance 348, b 

Westinghouse air-brakes 337 

Wheelbarrows, use in hauling earthwork 109. ^ 

Wheel resistances, train 343 



INDEX. 675 

Wheels and rails, mutual action and reaction 311-315 

effect of rigidly attaching them to axles 311 

White oak, use for trestles 129. 149 

ties 204 

Wire-drawn steam 321 

Wires and pipes, use in block signaling 309 

Wooden box culverts 188 

spikes, for filling spike holes 250 

YARDS AND TERMINALS.— Chap. XIII. 

Yards, engine 300 

freight 295-299 

grade in 295 

minor 297 

relation to main tracks 296 

transfer cranes 298 

track scales 299 

value of proper design 293 



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MAR 30 1903 



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